Variational calculus for hypersurface functionals: Singular Yamabe problem Willmore energies

Journal of Geometry and Physics 138 (2019) 168–193

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Variational calculus for hypersurface functionals: Singular Yamabe problem Willmore energies Michael Glaros c , A. Rod Gover a,b , Matthew Halbasch c , Andrew Waldron c ,

∗

a

Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand Mathematical Sciences Institute, Australian National University, ACT 0200, Australia c Department of Mathematics, University of California, Davis, CA95616, USA b

article

info

Article history: Received 21 November 2015 Received in revised form 19 December 2018 Accepted 29 December 2018 Available online 11 January 2019 Keywords: Calculus of variations Conformally compact Conformal geometry Hypersurfaces Willmore energy Yamabe problem

a b s t r a c t We develop an efficient calculus for varying hypersurface embeddings based on variations of hypersurface defining functions. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compact four-manifolds. We give explicit formulæ for both the energy functional and the obstruction. Vanishing of the latter is a necessary condition for solving the vacuum cosmological Einstein equations in four spacetime dimensions with data prescribed on a conformal infinity, while the energy functional generalizes the scheme-independent contribution to entanglement entropy across surfaces to hypersurfaces. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Hypersurface geometry is critically important for the analysis of boundary problems in mathematics and physics. Conformal hypersurfaces form a subtle but important class of cases, particularly because of their role in treating the boundary at infinity of conformally compact structures. Naturally the fundamental problem is the construction and study of local and global invariants. The work [15,17] develops an effective approach to hypersurface geometries by treating them as conformal infinities (see also the announcement [16]). This approach includes a calculus for hypersurface invariants based on defining functions. In this article we develop the corresponding variational theory; namely a hypersurface variational calculus based around defining functions. A main result is Theorem 3.11 which provides the general formula for computing variations of embeddings of hypersurface functionals by varying the underlying hypersurface defining function. A key motivation for our development of that theory was to develop an effective approach to computing the Euler–Lagrange equations corresponding to the new conformal energies of [15,17]. These functionals generalize to higher dimensions the Willmore energy [41], also known as the rigid string action [36]. For the crucial case of four manifolds, we show that the gradient of this functional agrees with the obstruction to smoothly solving the Yamabe problem for conformally compact structures. The Willmore energy and its associated gradient provide important invariants for surfaces. Indeed the Willmore conjecture concerning absolute minimizers of this energy has stimulated much geometric analysis [7,37]; note that Bryant’s work [7] includes a conformally invariant derivation of the Willmore gradient for surfaces embedded in S 3 . A general ∗ Corresponding author. E-mail addresses: [email protected] (M. Glaros), [email protected] (A.R. Gover), [email protected] (M. Halbasch), [email protected] (A. Waldron). https://doi.org/10.1016/j.geomphys.2018.12.018 0393-0440/© 2019 Elsevier B.V. All rights reserved.

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proof of the Willmore conjecture was provided by Marques and Neves [29,30]. In physics, the Willmore energy functional has applications in string theory where it appears in the entanglement entropy for a three dimensional conformal field theory [3,34]. It also arises as a certain log coefficient in a holographic notion of physical observables, and is closely related to the so-called conformal anomaly [23]. Furthermore it appears as the renormalized area of a minimal surface embedded in a hyperbolic three-manifold [1]. An important aspect of these invariants is their conformal invariance. While the Willmore energy integrand is not surprising its gradient, with respect to variations of embeddings, is an important local conformal invariant because it has linear leading term. This is discussed in some detail in [15,17]. We call this gradient the Willmore invariant. These applications and observations suggest a potentially significant role in geometry and physics for corresponding conformal invariants in higher dimensions. For four-dimensional hypersurfaces in Euclidean spaces, conformal energy functionals have been provided in [25,40] and [22]. Because it is the obstruction to smoothly solving the boundary Yamabe problem, vanishing of the three-dimensional hypersurface analog of the Willmore invariant given in Proposition 1.1 is a necessary condition for solving the four spacetime dimension vacuum, cosmological Einstein equations with negative constant curvature and prescribed boundary data along a conformal infinity. Moreover, its corresponding functional given in Eq. (1.1) is the three-dimensional hypersurface analog of the scheme-independent contribution to the entanglement entropy across surfaces [3,34]. Indeed it has been shown that for hypersurfaces of every dimension, there exists a conformally invariant analog of both the Willmore energy and the Willmore invariant. These are canonical in the sense that they are generated and constructed uniformly via a singular Yamabe problem that places the hypersurface as the boundary at infinity of a conformally compact, constant scalar curvature, manifold [15–17]. In each higher dimension, the analog of the Willmore invariant is called the obstruction density. This work was partly inspired by the results [2] of Andersson–Chruściel–Friedrich who studied boundary asymptotics for this Loewner–Nirenberg-type problem [28]. They found that the three-manifold obstruction to smoothness was a conformal invariant and also gave some information about the corresponding obstructions in higher dimensions. For three-manifold boundaries the obstruction density is easily shown to agree with the Willmore invariant. In [15,16] the singular Yamabe problem was recast as the tool for treating hypersurface conformal geometry that is analogous to the Fefferman–Graham Poincaré-metric as an approach to intrinsic conformal geometry. From this perspective, and using tractor calculus to re-express the Yamabe equation, the problem of computing the obstruction is reduced to a simple algorithm that also reveals the qualitative aspects of the obstruction densities and higher Willmore energies. In particular there is a dimension parity dichotomy: For each even hypersurface dimension, the obstruction density is a linear-leading-order conformal invariant that is a close analog of the Willmore invariant (and should be considered as a fundamental scalar curvature invariant); via elementary representation theory, odd dimensional hypersurfaces cannot admit linear leading order invariants. It is an interesting problem to understand the higher Willmore energies and obstruction densities for odd dimensional hypersurfaces. As explained above, a particularly important case for both mathematics and physics is when these are fourmanifold boundaries. In this article we employ the general holographic formula for the obstruction density [15] to compute the four-manifold obstruction density: Proposition 1.1. The obstruction density for d¯ = 3 is given by

] nˆ nˆ ) 1 [ ab( 2 ab 2 L 2II˚ (ab)◦ + F(ab)◦ − II˚ Bab + 12 K 2 + 2F (ab)◦ Fab + F (ab)◦ II˚ ab + W abc W⊤ abc . 6 Theorem 5.10 recalls that Bd¯ is a conformally invariant density of weight −d, and our tensor notations are explained in Section 1.1, while the hypersurface invariants appearing in the above formula are introduced in Section 2. In particular II˚ ab 2 denotes the trace-free second fundamental form, the conformally invariant Fialkow tensor Fab := II˚ ab + low er order terms ab (see Eq. (2.7)) and L is a second order, conformally invariant differential operator (that appears in Bernstein–Gel’fand– ¯ a ∇¯ b II˚ ab + lower order terms. Gel’fand (BGG) sequences) such that Lab II˚ ab = ∇ As mentioned above, another main result in [15,17] is the construction of conformally invariant functionals that generalize the Willmore energy to higher dimensions. For a hypersurface Σ of dimension d¯ these take the form B3 =

Id¯ = αd¯

∫ Σ

dAg¯ NA Pd¯ N A ,

where αd¯ is a non-vanishing constant and Pd¯ is a uniformly constructed family of conformally invariant operators, and N A d¯

is the normal tractor of [5] introduced in Section 5.2. For d¯ even, the operators Pd¯ take the form (∆⊤ ) 2 + low er order terms, so are hypersurface analogs of the GJMS operators of Graham, Jenne, Mason and Sparling [21]. For d¯ both even and odd a ab holographic formula for Pd¯ is known. This gives that P3 takes the form II˚ ∇a⊤ ∇b⊤ + low er order terms (the detailed formula ab appears in Eq. (6.2)). So for generic hypersurfaces P3 is Laplacian-like if one views II˚ as a (possibly indefinite) inverse metric. Using these results for P3 yields I3 =

1 6

∫ Σ

ab dAg¯ II˚ Fab ,

where the prefactor α3 =

1 6

has been chosen for later convenience.

(1.1)

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Our final main result is that the functional gradient of the energy I3 agrees precisely with the obstruction B3 to the singular Yamabe problem mentioned above: Proposition 1.2. With respect to hypersurface variations, the gradient of I3 is B3 . Our article is structured as follows: In Section 2, we review basic hypersurface theory and use conformal invariance as the organizing principle when producing expressions for hypersurface invariants (including a novel hypersurface Bach tensor Bab which already appears in Proposition 1.1). In Section 3 we develop our hypersurface variational calculus and treat Willmore and minimal surfaces as basic examples. In Section 4 we compute the variational gradient for our four-manifold analog of the Willmore energy. Section 5 is devoted to a review of the essential tractor calculus methods needed to effectively treat the singular Yamabe problem. In Section 6 we compute the obstruction density for four-manifold hypersurfaces. We also compute the obstruction for example metrics, including a hypersurface surrounding the horizon of a Schwarzschild black hole. The appendix gives explicit formulæ for the obstruction density for manifolds of dimensions five or less (so including B4 ) that can easily be implemented for explicit metrics using computer software packages. Graham has proven that the obstruction density is variational in all dimensions [20], and arises as the gradient of an action given by the coefficient of the log term in a renormalized volume expansion for the singular Yamabe problem treated in [15–17]. Furthermore, a holographic formula for the corresponding actions in all dimensions has been developed in [18,19]. More recently, Graham and Riechert have extended earlier work of Graham and Witten to perform a detailed study of Willmore-type functionals for embedded submanifolds [22]. 1.1. Riemannian and conformal geometry conventions We work with manifolds M of dimension d and embedded hypersurfaces of dimension d¯ := d − 1. When the dimension d equals three or four, we often refer to the latter as surfaces and spaces, respectively. (Note that the exterior derivative will be denoted by d, to avoid confusion with the dimension d.) When M is equipped with a Riemannian metric g, its Levi-Civita connection will be denoted by ∇ or ∇a in an abstract index notation (cf. [33]). The corresponding Riemann curvature tensor R is R(u, v )w = [∇u , ∇v ]w − ∇[u,v] w , for arbitrary vector fields u, v and w . In the abstract index notation, R is denoted by Rab c d and R(u, v )w is ua v b Rab c d w d . Cotangent and tangent spaces will be canonically identified using the metric tensor gab , so in other words the metric will be used to raise and lower indices in the standard fashion. The Riemann curvature can be decomposed into the trace-free Weyl curvature Wabcd and the symmetric Schouten tensor Pab according to Rabcd = Wabcd + 2ga[c Pd]b − 2gb[c Pd]a . Here antisymmetrization over a pair of indices is denoted by square brackets so that X[ab] := Ricci tensors are related by Ricbd := Rab a d = (d − 2)Pbd + gab J ,

1 2

(

)

Xab − Xba . The Schouten and

J := Paa .

The scalar curvature Sc = g ab Ricab , thus J = Sc/(2(d − 1)). In two dimensions the Schouten tensor is ill-defined, but we take J = 12 Sc in that case. In dimension d ≥ 3, the curl of the Schouten tensor gives the Cotton tensor Cabc := 2∇[a Pb]c . In dimensions d ≥ 4, (d − 3)Cabc = ∇d Wab d c .

(1.2)

A conformal geometry is a manifold equipped with a conformal structure c, namely an equivalence class of metrics with g′

g

equivalence relation g ′ ∼ g when g ′ = Ω 2 g for C ∞ M ∋ Ω > 0. The corresponding Weyl tensors obey W ab c d = W ab c d and thus the Weyl tensor is a conformal invariant. Conformal invariants may also be density-valued. A weight w density ]w [ (w ∈ R) is a section of the oriented line bundle (∧d TM)2 2d =: E M [w]. Vector bundles V M tensored with this are denoted V M ⊗ E M [w] =: V M [w]. Each metric g ∈ c determines a section of (∧d T ∗ M)2 and so there is a tautological global section g of ⊙2 T ∗ M [2] called the conformal metric. Hence, each g ∈ c is in 1 : 1 correspondence with a strictly positive section τ ∈ E M [1] via g = τ −2 g . Such sections τ will be referred to as a true scale, and we will often present conformally invariant formulæ in terms of a Riemannian metric g by making an (arbitrary) choice of scale τ . It will be clear from context where this has been done. In situations where we are dealing with tensors built from higher rank tensors by contractions with vectors or covectors u

into the first slot such as X (u, ·) (or equivalently ua Xab ), we will often use the notation X a . In higher rank cases where more u

indices are missing, an alternating leftmost–rightmost contraction procedure has been followed, so if X is rank 3, then X b denotes X (u, ·, u) or equivalently ua Xabc uc . Contractions across multiple indices of tensors with one another will be denoted

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by an obvious bracket notation, for example, if Y is rank two and X is rank three, X (Y , ·) denotes the covector Xabc Y ab . Also, for 2 rank two symmetric tensors S , T , we will have occasion to employ matrix notations such as Sab := Sac Sbc and tr ST := Sab T ba . Unit weight symmetrization over groups of indices is denoted by round brackets, which we adorn with the symbol ◦ when ( ) projecting onto the trace-free part of this, for example X(ab)◦ := 21 Xab + Xba − 1d gab Xc c ; for mixed symmetry tensors, projection onto the trace-free part of a group of indices will be indicated by the notation { }◦ and their corresponding section spaces will be decorated with a subscript ◦. Finally, we will also use a√dot product √ notation for inner products of vectors g(u, v ) = ua v a = u.v as well as divergences ∇.Xb := ∇ a Xab , and |u| := ua ua := u2 denotes the length of a vector u. 2. Embedded hypersurfaces A hypersurface Σ is a smoothly embedded codimension 1 submanifold of a smooth manifold M. A function s ∈ C ∞ M is called a defining function for Σ if the hypersurface is given by its zero locus, i.e. Σ = Z (s) and the exterior ds =: n ( derivative )⏐ is nowhere vanishing along Σ . The covector field n|Σ is then a conormal to the hypersurface and nˆ a := na /|n| ⏐Σ is the unit conormal while na |Σ is a normal vector. The conormal allows the tangent bundle T Σ and the subbundle of TM |Σ orthogonal to na , denoted TM ⊤ , to be canonically identified. Thus we may use the same abstract indices for T Σ as for TM. The projection of tensors, to hypersurface tensors will be denoted by a superscript ⊤. Given an extension nˆ a of the unit normal to M, we will also write v ⊤ := v − nˆ nˆ .v for vectors (and similarly for tensors) v ∈ TM. Upon restriction, these give hypersurface vectors (and tensors). The metric g on M induces a metric g¯ on Σ given by g¯ab = gab |Σ −ˆna nˆ b .

(2.1)

Given a hypersurface vector field v ∈ Γ (T Σ ), the Gauß formula relates the restriction to T Σ of the Levi-Civita connection ∇ on M to the hypersurface Levi-Civita connection of g¯ according to a

∇¯ a v b = ∇a⊤ v b + nˆ b IIac v c .

(2.2)

In the above formula, the right hand side is defined independently of how v a ∈ T Σ is extended to v a ∈ TM. In general, we will use the term tangential for operators O along Σ if they have the property that for v a smooth extension of some tensor v¯ defined along Σ , the quantity Ov|Σ is independent of the choice of this extension. In the above display, the second fundamental form IIab ∈ Γ (⊙2 T ∗ Σ ) is given by IIab = ∇a⊤ nˆ b . Its ‘‘averaged’’ trace is the mean curvature H=

1 d¯

IIaa .

We will often use bars to distinguish intrinsic hypersurface Σ quantities from their host space counterparts. In particular, the equations of Gauß relate ambient curvatures to intrinsic ones according to

¯ R⊤ abcd = Rabcd − 2IIa[c IId]b , 2 ¯ Ric⊤ ab = Ricab + IIab − dHIIab ,

Sc − 2 Ric(nˆ , nˆ ) = Sc + tr II2 − d¯ 2 H 2 .

(2.3)

The last of these recovers Gauß’ Theorema Egregium for a Euclidean ambient space. The Codazzi–Mainardi equation gives the covariant curl of the second fundamental form in terms of ambient curvature 1 nˆ ∇¯ [a IIb]c = R ⊤ cba .

2 Finally the divergence of the Codazzi–Mainardi equation determines the Laplacian of mean curvature:

¯H = ∆

1( d¯

nˆ ) ¯ ∇. ¯ II − ∇¯ a Ric⊤a . ∇.

(2.4)

(2.5)

2.1. Conformal hypersurfaces The ambient conformal structure c induces a conformal structure c¯ on Σ . Locally we may always assume there is a section nˆ a ∈ Γ (T ∗ M [1]) obeying g ab nˆ a nˆ b = 1. This conformal unit conormal plays the role of a Riemannian unit conormal. The trace-free part of the second fundamental form II˚ ab = IIab − H g¯ab , defined (in a choice of scale) in terms of nˆ a via Eq. (2.2), is a conformally invariant section of ⊙2 T ∗ M [1]. Thus the quantity 2

K := tr II˚ ∈ Γ (E Σ [−2]) , is a conformal density of weight −2, that we call the rigidity density.

(2.6)

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Conformal invariance of the trace-free second fundamental form II˚ and the Weyl tensor can be used to decompose the equations of Gauß and Codazzi–Mainardi equations into conformally invariant parts. First, the Ricci equation in (2.3) gives what we shall call the Fialkow–Gauß equation (cf. [40]) 2 II˚ ab −

cd ( ) nˆ 1 II˚ cd II˚ − P¯ ab + H II˚ ab + g¯ab H 2 =: (d¯ − 2)Fab . g¯ab ¯ − W ab = (d¯ − 2) P⊤ ab d−1 2 2

1

(2.7)

The left hand side of this equation is conformally invariant, which proves that the Fialkow tensor Fab defined by the above in dimension d ≥ 4 is a conformally invariant section of ⊙2 T ∗ M [0]. This combined with the trace-free second fundamental form gives a new distinguished density L := tr(II˚ F ) ∈ Γ (E Σ [−3]) ,

(2.8)

which – in analogy with the Willmore energy’s rôle as an action principle for a rigid string [36] – we call the membrane rigidity density. In dimensions d ≥ 4, the trace-free part of the first Gauß equation allows the tangential part of the ambient Weyl tensor to be traded for its hypersurface counterpart: ⊤ ¯ abcd − 2II˚ II˚ − 2g¯ F + 2g¯ F . =W Wabcd a[c d]b a[c d]b b[c d]a

(2.9)

In dimensions d ≥ 4, the third Gauß equation expresses the rigidity density in terms of the difference between ambient and hypersurface scalar curvatures:

(

K = 2(d¯ − 1) J − P(nˆ , nˆ ) − J¯ +

d¯ 2

)

H2 .

d¯ 2

This shows that J − P(nˆ , nˆ ) − J¯ + H 2 is a conformally invariant weight −2 density. The trace-free part of the Codazzi–Mainardi equation shows that the trace-free curl of the trace-free second fundamental form is a weight 1 conformal invariant: n

¯ ˚ W⊤ abc = 2∇{[c IIb]a}◦ .

(2.10)

Finally, we will need a certain novel four-manifold hypersurface invariant: Lemma 2.1. Let d¯ = 3 and nˆ

nˆ

nˆ

¯c ⊤ Bab := C ⊤ (ab) + H W ab − ∇ W (ab)c . g

g

Then Bab defines a symmetric, weight −1 conformal hypersurface density Bab ∈ Γ (⊙2 T ∗ Σ [−1]) g

given in a choice of scale g ∈ c by Bab . Proof. This result can be established by examining the conformal transformation properties of the Cotton tensor, mean curvature and the divergence of the Weyl tensor: Ω 2g

Cabc = Ω −1 (Cabc − Wabc d Υd )⊤ ,

HΩ

2g

= Ω −1 (H + Υ .ˆn),

nˆ nˆ nˆ [ c nˆ ⊤ ]Ω 2 g ( ) c ⊤ c ⊤ ¯ ¯ ∇¯ W abc = Ω −1 ∇¯ c W ⊤ abc + (d − 4)Υ W (ab)c + (d − 2) Υ W [ab]c ,

(2.11)

where Υa := Ω −1 ∇a Ω . For the Cotton tensor transformation we must decompose Υd = Υd⊤ + nˆ d Υ .ˆn, the second term of Ω2g

which cancels the term in Bab produced by the mean curvature. Setting d¯ = 3 and keeping only the symmetric part of the Weyl divergence terms gives the final cancellation. □ Remark 2.2. Invariance of the density Bab for almost Einstein structures follows from [12, Proposition 4.3], where it was related to the Bach tensor. It also appeared in the construction of a tractor analog of the exterior derivative in [13]. Hence we shall call Bab the hypersurface Bach tensor. ⋆ 2.1.1. Invariant operators Conformally invariant operators are maps between sections of conformally weighted tensor bundles. These play a fundamental role in conformal and hypersurface geometries. Quite generally we will be interested in examples of these whose target and domain have differing weights and possibly tensor types. There are two main examples that are crucial to following developments.

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

173

Proposition 2.3. Let d ≥ 3 and X ab be a rank 2, symmetric, weight −d = −d¯ − 1, trace-free hypersurface tensor. The mapping, given in a choice of scale by X ab ↦ −→ Lab X ab :=

{

∇¯ a ∇¯ b X ab + P¯ ab X ab , ⊤ ab ∇¯ a ∇¯ b X ab + Pab X + H II˚ ab X ab ,

d¯ ≥ 3 , d¯ = 2 ,

(2.12)

determines a conformally invariant operator mapping Lab

Γ (⊙2◦ T Σ [−d ]) −→ Γ (E Σ [−d ]) . Proof. In dimensions d¯ ≥ 3, the result is standard and germane to any conformal geometry from the theory of invariant differential operators based on BGG sequences, see for example [9]. When d¯ = 2 one can choose a scale, and transform the metric explicitly under gab → Ω 2 gab and then directly verify conformal invariance using the mean curvature transformation given in Eq. (2.11) as well as those for the Schouten tensor and a weight w vector v a (which extends to higher rank tensors by linearity): 1

Ω 2g

= Pgab − ∇a Υb + Υa Υb − Υ 2 gab , 2 ( ) 2 ∇bΩ g v a = Ω w ∇bg v a + (w + 1)Υb v a − Υ a vb + δba Υ .v . □ Pab

2

Observe that Lab is well-defined when acting on the conformal density II˚ (ab)◦ ; we record an identity for this required in future developments: Proposition 2.4. Along hypersurfaces in a four-manifold one has 2

Lab II˚ (ab)◦ =

1 ( 3

nˆ

nˆ

nˆ

bc 1 ¯ II˚ a )(∇. ¯ II˚ a ) + 2 II˚ ab∆ ¯ II˚ ab − 2 J¯ K − 4 II˚ ab ∇¯ c W⊤ W abc W⊤ ∇¯ a II˚ )(∇¯ a II˚ bc ) + 12 (∇. abc + 2 abc . 3 3 3

Proof. The proof is a combination of basic tensor algebra and repeated usage of the trace-free Codazzi–Mainardi Equa2 tion (2.10) applied to the double divergence of II˚ (ab)◦ . We record the main steps below: 1 2 c ¯K ∇¯ a ∇¯ b II˚ (ab)◦ = 2∇¯ a (II˚ c (a ∇¯ b II˚ b) ) − ∆ 3

¯ II˚ a )(∇. ¯ II˚ ) + 2II˚ ∇¯ ∇¯ a II˚ bc + (∇¯ a II˚ bc )(∇¯ b II˚ ac ) + II˚ ac [∇¯ a ,∇¯ b ]II˚ bc − 1 ∆ ¯K = (∇. a

ab

c

3

1

1

a ˚ bc

a

2

ab

a ˚ bc

nˆ

¯ II˚ a )(∇. ¯ II˚ ) − II˚ ∆ ¯ II˚ ab + (∇¯ II )W cab = (∇¯ II )(∇¯ a II˚ bc ) + (∇. 3

2

⊤

3

ac b ab + 2II˚ ∇¯ c ∇¯ a II˚ bc + II˚ [∇¯ a ,∇¯ b ]II˚ c nˆ 1 1 nˆ 1 bc ¯ II˚ a )(∇. ¯ II˚ a ) + 2 II˚ ab ∆ ¯ II˚ ab + W cab W ⊤ = (∇¯ a II˚ )(∇¯ a II˚ bc ) + (∇. abc

3

−

2

4

nˆ

ab c ¯ W⊤ II˚ ∇ abc +

1

3

2

ac ¯ a ,∇¯ b ]II˚ bc . II˚ [∇

3 3 The third line was achieved using identity ab ¯ a ∇. ¯ II˚ b = II˚ ∇

2 3

ab ¯ II˚ ab − II˚ ∆

2 3

2 ¯ − J¯ K − 2 tr(II˚ P)

2 3

n

ab c ¯ W⊤ II˚ ∇ bac ,

(2.13)

¯ c ∇¯ a II˚ bc = ∆ ¯ II˚ a + lower order terms. In three dimensions, ¯ II˚ ab − ∇¯ b ∇. which can be proved by using Eq. (2.10) to rewrite ∇ ¯ a ∇¯ b II˚ 2(ab)◦ displayed above Eq. (2.13) can only contribute the Weyl tensor vanishes so the last term in the computation of ∇ 2 intrinsic Schouten and scalar curvature terms; these are precisely −P¯ ab II˚ − 2 J¯ K and this completes the proof. □ (ab)◦

3

The second conformally invariant hypersurface operator needed implements the Robin combination of Neumann and Dirichlet boundary conditions and was shown to be conformally invariant by Cherrier [8]: Definition 2.5. Given a weight w conformal density f ∈ Γ (E M [w]), we define the (conformal) Robin operator

⏐ ⏐ δnˆ f := ∇nˆ f ⏐Σ − wH · f ⏐Σ .

(2.14)

w Remark 2.6. The operation (∇nˆ − d− ∇.ˆn)f is the Lie derivative of the density f along the unit normal vector nˆ , so by 1 construction δnˆ f is a weight w − 1 conformal hypersurface density. ⋆

The Robin operator can be extended to also act on conformal tensor densities, in what follows we will need the case of a rank two, symmetric conformal tensor density:

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Lemma 2.7. Suppose Xab ∈ Γ (⊙2 T ∗ M [w]) obeys nˆ .Xb |Σ = 0 then one has a conformally invariant operator given here in a scale g ∈c

( )⊤ ⏐ δnˆ Xab := ∇nˆ Xab − (w − 2)HXab ⏐Σ .

(2.15)

This defines a section of Γ (⊙2 T ∗ Σ [w − 1]) in the corresponding induced scale g¯ ∈ c¯ . Proof. This is a member of a general class of results that can be obtained from the Laplace–Robin operator discussed in Section 5.5. A simple proof of the statement uses the conformal transformation property of the mean curvature given in Eq. (2.11) and that for the normal derivative of a rank two tensor density:

) ( 2 ∇nˆΩ g Xab = Ω w−1 ∇nˆ Xab + (w − 2)Υ .ˆnXab − 2Υ(a nˆ .Xb) + 2nˆ (a Υ .Xb) . Since nˆ .Xb |Σ = 0, and the formula for δnˆ projects onto the tangential part of ∇nˆ Xab , only the first two terms on the right hand side above survive, and the result thus follows. □ 3. Defining function variational calculus To handle hypersurface variational problems we treat hypersurfaces in terms of defining functions. Locally any hypersurface is the zero locus of some defining function, so no generality is lost studying hypersurfaces Σ = Z (s) for some defining function s. For simplicity we take M oriented with volume form ω; this and the conormal ds determine the orientation of Σ . We also shall assume, again mostly for simplicity, that Σ is compact. Hypersurface invariants are conveniently defined as follows (see [15, Definition 2.6], cf. [10]): Definition 3.1. A scalar Riemannian preinvariant is a mapping P which assigns a smooth function P (s; g) to each Riemannian d-manifold (M , g) and hypersurface defining function s, satisfying: (i) P (s; g) is natural; for any diffeomorphism φ : M → M we have P (φ ∗ s, φ ∗ g) = φ ∗ P (s; g) .

(ii) If s′ = v s for some smooth, positive function v then P (s; g)|Σ = P (v s; g)|Σ , where Σ = Z (s) .

(iii) P is given by a universal polynomial expression such that, given a local coordinate system (xa ) on (M , g), P (s; g) is given by a polynomial in the variables gab , ∂a1 gbc , . . . , ∂a1 ∂a2 · · · ∂ak gbc , (det g)−1 , ωa1 ...ad , 1 s, ∂b1 s, . . . , ∂b1 ∂b2 · · · ∂bℓ s, |ds|− g ,

for some positive integers k, ℓ. Here ∂a means ∂/∂ xa , gab = g(∂a , ∂b ), det g = det(gab ) and ωa1 ...ad = ω(∂a1 , . . . , ∂ad ). A scalar Riemannian invariant of a hypersurface Σ is the restriction P¯ (Σ ; g) := P (s; g)|Σ of a preinvariant P (s; g) to Σ . Remark 3.2. A hypersurface invariant P¯ depends only on the data (M , g , Σ ). For point (i) note that if Σ = Z (s), then φ −1 (Σ ) is a hypersurface with defining function φ ∗ s. In (ii), the requirement s′ = v s means that s′ and s are two compatibly oriented defining functions with Z (s) = Z (s′ ) =: Σ . In (iii), since we are ultimately interested in hypersurface invariants, there is no 1 loss of generality studying defining functions such that |ds|− ̸= 0 everywhere in M, since we may, if necessary replace M g by a local neighborhood of Σ . The definition extends to tensor-valued hypersurface preinvariants and invariants by considering tensor-valued functions P and requiring the coordinate components of the image satisfy conditions (ii), (iii), and the obvious adjustment of (i). The term hypersurface invariant will be taken to mean either tensor or scalar valued hypersurface invariants. ⋆ Local Riemannian invariants, constructed in the usual way, provide trivial examples of hypersurface preinvariants. Examples of preinvariants that depend non-trivially on both s and g are given below: Example 3.3. Given a hypersurface defining function s, examples of preinvariants are Pa (s; g) =

∇a s and Pab (s; g) = ∇a |∇ s|

(

∇b s |∇ s|

) −

∇a s ∇ c s ∇c |∇ s| |∇ s|

(

∇b s |∇ s|

)

.

Upon restriction to Σ = Z (s), these give the unit conormal and second fundamental form hypersurface invariants. It is important to note that for distinguished choices of defining function s, these expressions simplify considerably. For example, if s obeys |∇ s| = 1, then the second fundamental form is IIab = (∇a ∇b s)|Σ , but the quantity ∇a ∇b s is, of course, not a preinvariant. ⋆

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

175

∫

We now consider integrals Σ dAg¯ • over a hypersurface, where dAg¯ is the volume element of the induced metric g¯ab . Let s be a defining function for Σ . Then, since (see Eq. (2.1)) ds ⊗ ds )⏐⏐

(

g¯ = g −

⏐ ,

|ds|2

Σ

choosing coordinates (x ) = (s, yi ) where g −1 (ds, dyi )|Σ = 0, it follows that a

√

√

⏐ ⏐

det g ⏐

Σ

det g¯ ⏐ . |ds|⏐Σ

=

Thus we have the following: Proposition 3.4. Let Σ be a smoothly embedded, compact hypersurface in M, with defining function s, and let P¯ be a hypersurface invariant for Σ with preinvariant P (s; g). Then

∫ Σ

dAg¯ P¯ =

∫

dVg |∇ s| δ(s) P (s; g) ,

(3.1)

M

where g¯ is the pullback of the metric g to Σ . In the above δ(s) is the Dirac delta distribution. The above formula is standard, see for example [32, Section 1.5] for its use in the theory of implicit surfaces. In particular, we use that the defining function s can be used as a coordinate in a cylinder neighborhood of the hypersurface Σ . Then the Dirac delta distribution of the defining function s is defined with respect to smooth test functions that are compactly supported in this variable (which may be taken as a coordinate in a neighborhood of Σ ). In fact, to make the right hand side Eq. (3.1) well-defined, a smooth cut-off function is introduced in an obvious way as described in [18, Section 2.8]. This is independent of the choice of cut-off function. This distribution-based approach has the advantage of allowing us to perform surface integrals over hypersurface invariants P¯ in terms of (often far simpler) extensions to M. This approach necessitates handling derivatives of Dirac delta distributions. For that δ′ (s) is interpreted in the usual distributional sense; namely s is taken as a coordinate and δ′ (s)f (s) = −δ(s)f ′ (s) (under the integral) for any test function f (s). It follows that the gradient of δ(s) satisfies the distributional identity

∇δ(s) = (∇ s) δ′ (s) .

(3.2)

Remark 3.5. Note that the distributional identities

∑ δ(x − x0 ) |f ′ (x0 )|

xδ(x) = 0 and δ(f (x)) =

x0 ∈Z (f )

for functions f with simple roots, imply |∇ (v s)|δ(v s) = |∇ s|δ(s), where v is any smooth positive function and once again s is viewed as coordinate in a neighborhood of Σ , as explained above. Thus, the integrand |∇ s|δ(s)P (s; g) on the right hand side of (3.1) can be loosely viewed as distributional hypersurface preinvariant. ⋆

∫ Moving now to variational considerations, let P¯ be a hypersurface invariant. The variation of the functional I(Σ ) = dAg¯ P¯ , with respect to the embedding of the hypersurface Σ , is defined by Σ ∫ ⏐ ⏐ d dI(Σt ) ⏐ ⏐ δ I := dAg¯t P¯ (Σt ; g) ⏐ . (3.3) ⏐ = dt

t =0

dt

t =0

Σt

Here Σt denotes a smooth, one parameter family of hypersurfaces such that Σ0 = Σ and Σt = Σ outside a compactly supported subset of M. Also, g¯t is the pullback of the metric g to Σt . Our strategy is, without loss of generality, to consider a family of hypersurfaces Σt = Z (st ) where st is the family of defining functions st := s + tu

(3.4)

∞ for some smooth, (compactly )⏐ supported δ s := u ∈ C M. Thus I(Σt ) = I(Z (st )). In general we will use δ to denote the df (s+tu) ⏐ operation f (s) ↦ → =: δ f . dt t =0 Using Eq. (3.1) the variational formula (3.3) becomes

δI =

d dt

∫

⏐ ⏐

M

dVg |∇ st | δ(st )P (st ; g) ⏐

t =0

.

This reformulation leads to the following Lemma which is needed for our key variational Theorem 3.11, and is also a useful computational tool:

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Lemma 3.6. Let P be a hypersurface preinvariant with corresponding hypersurface invariant P¯ . Consider a compactly supported variation δ s of a defining function s for a hypersurface Σ . Then

)

(∫

δ

dAg¯ P¯

Σ

∫

[

dVg |∇ s|δ(s) δ P −

= M

] δs (∇nˆ + ∇.ˆn)P , |∇ s|

(3.5)

where nˆ := (∇ s)/|∇ s|. Proof. Let χ be a (fixed) compactly supported cutoff function taking the value 1 on an open neighborhood of the support of δ s. Then

δ

)

(∫ Σ

dAg¯ P¯

)

(∫

dVg χ |∇ s| δ(s) P =δ ∫ M [ ] = dVg χ (∇nˆ δ s)δ(s) P + |∇ s| δ′ (s) δ s P + |∇ s| δ(s) δ P ∫M [ ] = dVg χ −(∇.ˆn)δ s δ(s) P − δ(s) δ s ∇nˆ P + |∇ s| δ(s) δ P .

(3.6)

M

Here we used that the derivative of χ vanishes where δ s has support and ∇nˆ δ(s) = |∇ s|δ′ (s) (see Eq. (3.2)). In the second line above we performed an integration by parts using that the integrand has compact support. The result follows using Eq. (3.1). □

¯ the integral Remark 3.7. When P (and thus also P¯ ) is a density of weight −d, dAg¯ P¯ is conformally invariant. In this Σ case the operator appearing in the second term of Eq. (3.5) is the Lie derivative of P along nˆ . Along Σ this becomes

∫

(

)⏐ ⏐

(∇nˆ + ∇.ˆn)P ⏐

Σ

= δnˆ P ,

where δnˆ is Cherrier’s conformally invariant Robin operator; see Definition 2.5. This motivates the notation of the definition that follows. ⋆ Definition 3.8. Let Σ be a hypersurface and let f ∈ C ∞ M (or a density of any weight). Then we define

⏐ δk f = ∇nˆ f ⏐Σ − k H f¯ , where f¯ = f |Σ and k ∈ C.

∫

To compute the variational gradient of Σ dAg¯ P¯ , it remains to express the first term on the right hand side of Eq. (3.5) in ∫ the form M dVg |∇ s|δ(s) δˆ s P ′ with some local formula for P ′ where

δˆ s :=

δs . |∇ s|

We have introduced the quantity δˆ s for homogeneity reasons: Observe that the integral Σ dAg¯ P¯ being differentiated on t the left hand side of (3.5) is independent of(∫ the defining ) function st , so in particular is unchanged upon replacing st with v st for any positive, smooth function v . Thus δ Σ dAg¯ P¯ =: δ I(s, δ s) = δ I(v s, vδ s). It is easy to verify the identity

∫

⏐ ⏐ δ s ⏐⏐ vδ s ⏐⏐ = , |∇ (v s)| ⏐Σ |∇ (s)| ⏐Σ so that along Σ , δˆ s enjoys the same homogeneity property as δ I. To rewrite δ I with δˆ s undifferentiated, we will need the following ‘‘integration by parts’’ lemma: Lemma 3.9. Let s be a defining function and suppose C a ∈ Γ (TM). Moreover let B be a test function (i.e., smooth with compact support). Then

∫

dVg |∇ s|δ(s) C a ∇a⊤ B = −

∫

M

dVg |∇ s|δ(s)B ∇a⊤ C ⊤ a , M

where ∇ ⊤ = ∇ − nˆ ∇nˆ , nˆ a = (∇a s)/|∇ s| and C ⊤ := C − nˆ nˆ .C . Proof. First, not forgetting the dependence of ∇ ⊤ on nˆ , we use that B is a test function and integrate by parts to compute

∫

dVg |∇ s|δ(s) C a ∇a⊤ B = − M

∫

[

M

Next, a straightforward computation shows that

∇nˆ nˆ a =

∇a⊤ |∇ s| . |∇ s|

]

dVg B ∇a⊤ (|∇ s| δ(s)C ⊤ a ) − |∇ s| δ(s)C ⊤ b ∇a⊤ (nˆ a nˆ b ) .

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

177

Using Eq. (3.2) we have the distributional identity ∇ ⊤ δ(s) = (∇ ⊤s)δ′ (s), and so ∇ ⊤ δ(s) = 0 because ∇ ⊤s = 0. From this the result follows. □ Remark 3.10. When C a and B, respectively, take values in sections of some tensor bundle over M and its dual and these are appropriately contracted to form a scalar, the natural extension of the above Lemma follows immediately. ⋆ The variation δ P of a preinvariant is linear in the variation u = δ s and its derivatives up to for some finite order k. If all derivatives on the variation u are tangential derivatives ∇ ⊤ u, (∇ ⊤ )2 u, . . . , (∇ ⊤ )k u, we can use the above lemma to integrate these by parts. In fact, as we shall prove in Theorem 3.11, this is always the case. That is, we show there is a well-defined of a preinvariant determined by notion for the functional derivative δδP s

∫

dVg |∇ s| δ(s) δ P := M

∫

dVg |∇ s| δ(s) δˆ s M

δP , δs

(3.7)

for compactly supported variations δ s. We now give our main variational theorem: Theorem 3.11. Let Σ be a hypersurface with defining function s and let P¯ be a hypersurface invariant with corresponding preinvariant P . Then for compactly supported variations,

δ

∫ Σ

dAg¯ P¯ =

∫

dVg |∇ s|δ(s) δˆ s M

( δP δs

) − (∇nˆ + ∇.ˆn)P ,

where nˆ = (∇ s)/|∇ s|. Proof. Thanks to Lemma 3.6, we need only verify the well definedness of δδP in Eq. (3.7). Our first step is to construct s coordinates (s, xi ) where i = 1, . . . , d¯ such that the vector field ∂∂s is orthogonal to constant s = ε hypersurfaces Σ ε := Z (s − ε ). This is achieved by integrating the vector fields g −1 (ds, ·) and using the flow along their integral curves to push forward coordinates xi on Σ = Σ ε=0 to Σ ε . Since δ s has compact support, we may write the variation δ s = u ∈ C ∞ M in the coordinates (s, xi ) as u(s, xi ) = u(0, xi ) + s U(s, xi ) , for some smooth, bounded, function U. Now we decompose the one parameter family of defining functions in Eq. (3.4) according to st = s′ + tu0 , where the function u0 is defined in these coordinates by u0 = u(0, xi ). Now note that 1 + t U(s, xi ) is a positive function ′ i for t small, ∫ because U is bounded. (So s = s(1 + t U(s, x )) is also a defining function.) Thus for all such t we have that I(Σ ) := Σ dAg¯ P¯ satisfies I(Σ ) = I(Z (s)) = I(Z (s′ )) . Now from the construction of I it follows that I(t1 , tt ) := I(Z (s + t1 sU + t2 u0 )) is differentiable as a function on R2 . From this and the previous observation it follows that

δI =

dI(s + tu0 ) ⏐ dt

⏐ ⏐

t =0

.

Next, from the definition of a preinvariant it follows that the variation δ I is a sum of terms of the form

∫

dVg |∇ s|δ(s) C a1 ...ak ∇a1 · · · ∇ak u , M

for some smooth tensors C a1 ···ak (s; g). However, the above coordinate argument shows that we may replace this by

∫

dVg |∇ s|δ(s) C a1 ...ak ∇a1 · · · ∇ak u0 . M

However, since ∇nˆ u0 = 0, we have ∇ u0 = (∇ − nˆ ∇nˆ )u0 =: ∇ ⊤ u0 . An easy induction shows that we can reexpress the integral as a sum of terms

∫ M

dVg |∇ s|δ(s) C˜ a1 ...aℓ ∇a⊤1 · · · ∇a⊤ℓ u ,

where we written u not u0 because ∇ ⊤ s = 0 and sδ(s) = 0 (and C˜ a1 ...aℓ are some smooth tensors). Integrating all ∇ ⊤ ’s by parts according to Lemma 3.9 then establishes Eq. (3.7). □

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Corollary 3.12. Let Σ be a hypersurface with defining function s and let P¯ be a hypersurface invariant with corresponding preinvariant P . Then

δ P ⏐⏐ δP − (∇nˆ + ∇.ˆn)P ⏐ − δ-d¯ P and δs Σ δs are a hypersurface invariant and its corresponding preinvariant. Example 3.13. Let nˆ a = ∇a s/|∇ s| where s is a defining function for Σ . Then P (s; g) = ∇ a nˆ a

is a preinvariant for d¯ times the mean curvature. Noting that

δ nˆ a =

∇a⊤ δ s , |∇ s|

it follows that

δP = ∇ a

( ∇ ⊤ δs ) a

|∇ s|

⏐ ⏐2 ] [ = ∆⊤ − ⏐∇a⊤ log |∇ s|⏐ + (∆⊤ log |∇ s|) δˆ s ,

where ∇a⊤ := ∇a − nˆ a ∇nˆ and ∆⊤ := g ab ∇a⊤ ∇b⊤ . Notice δ P does not contain normal derivatives ∇nˆ acting on δˆ s. Thus

⏐ ⏐2 δP = ∆⊤ log |∇ s| − ⏐∇a⊤ log |∇ s|⏐ . δs This is not a preinvariant, but it is not difficult to verify that invariant

δP δs

− (∇nˆ + ∇.ˆn)P is and along Σ this equals the hypersurface

− tr II2 + d¯ 2 H 2 + Ric(nˆ , nˆ ) = Sc − Ric(nˆ , nˆ ) − Sc . Here the last equality relied on Eq. (2.3). ⋆ 3.1. Variational methods and identities We now explain our strategy for computing variations and collect some key identities. Using Theorem 3.11, hypersurface variations are given in terms of preinvariants. To express preinvariants in terms of standard hypersurface invariants, we take advantage of property (ii) of Definition 3.1 which allows us to choose any defining function when evaluating preinvariants along Σ . In a computational context we are often concerned with expressions built from a sum of terms which separately are not preinvariants even though their sum is. Therefore in calculations it is important to use the same choice of defining function for every term. Given any hypersurface in a Riemannian manifold there is locally a defining function s satisfying

|∇ s| = 1 . Defining functions with this property are dubbed unit defining functions. Since unit defining functions are determined by the hypersurface embedding, they provide a good choice for computations. In fact, there is a recursive solution to the formal problem of finding a smooth unit defining function s1 given a defining function s for a hypersurface such that

|∇ s1 | = 1 + sℓ f , for ℓ ∈ Z≥1 arbitrarily large and f ∈ C ∞ M (see [15, Section 2.2]). This explicitly relates the jets of s to hypersurface invariants. Σ

1

We employ the notation = for equalities relying on this choice while = is used when evaluating quantities along Σ . In general we abuse notation by using the same symbols for preinvariants as for their corresponding hypersurface invariant when the former has been explicitly chosen. Let us tabulate some commonly used choices for preinvariants as well as their expressions in terms of a unit defining function: 1

=

Preinvariant

Invariant

na := ∇a s

∇a s |∇ s|

nˆ a

gab − na nb

γab := gab − nˆ a nˆ b

g¯ab

∇ a nb

∇a⊤ nˆ b

IIab

1 d−1

∇a n

∇ a nb −

a

1 d−1

(gab −na nb )∇c nc d−1

⊤ ˆa

∇a n

∇a nb − γab H ⊤ˆ

H II˚ ab

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

179

1 So for example, off the hypersurface the notation nˆ a means (∇a s)/|∇ s| and an expression like ∇n H denotes d− ∇ ∇ ⊤ nˆ a . The 1 n a ⊤ label ⊤ is employed to denote projection of tensors on M to their part perpendicular to the unit vector nˆ , and ∇ := ∇ − nˆ ∇nˆ . Our first key variational identity was given in Example 3.13, but we repeat it for completeness:

δ nˆ a =

na ∇ a δs 1 Σ − 2 ∇nˆ δ s = ∇a⊤ δ s = ∇¯ a δ s| , | n| |n|

(3.8)

1 where δ s|:= δ s|Σ = δˆ s|Σ =: δˆ s|. Thus

]⊤ ( )⊤ Σ ( )⊤ [ ( ∇ ⊤ δ s ) ∇a⊤ δ s 1 − ∇nˆ nˆ b = ∇a⊤ ∇b⊤ δ s = ∇¯ a ∇¯ b δ s| . δ IIab = ∇a⊤ b |n| |n|

(3.9)

Importantly, in the above computations, we first varied preinvariant expressions for invariants and only thereafter specialized these to a unit defining function. In a functional setting, we can use Eq. (3.9) to give a useful lemma for computing variations of functionals involving the second fundamental form: Lemma 3.14. Let K ab ∈ Γ (⊙2 T Σ ) and s be a unit defining function with δ s|= δ s|Σ where δ s has compact support. Then

∫

1 dAg¯ K ab δ IIab ⏐Σ =

(

Σ

)⏐

∫ Σ

¯ a ∇¯ b K ab . dAg¯ δ s| ∇

(3.10)

Finally, one more technical lemma is required: Lemma 3.15. Let Xab be a rank two, symmetric tensor on M satisfying nˆ a Xab = 0. Then

)]⊤ ( [ = 0. ∇nˆ Xab − γab γ cd ∇nˆ Xcd − ∇nˆ X(ab)◦ We leave the proof to the reader. 3.2. Examples As a first example, we consider the problem of extremizing the area of an embedded hypersurface. The relevant functional is

∫ I= Σ

dAg¯ .

Using Theorem 3.11, we have

δI = −

∫ Σ

dAg¯ δˆ s| (δ-d¯ 1) = − d¯ ·

∫ Σ

dAg¯ δˆ s| H ,

which recovers the standard vanishing mean curvature H condition for minimal surfaces. A second example is the variation of the Willmore energy for surfaces (in Lorentzian signature, this is the rigid string action of [36]): I2 = −

1

∫

6

Σ

dAg¯ K ,

(3.11)

where K is the rigidity density of Eq. (2.6). Using Lemma 3.6, we find 1

δ I2 = −

1 6

∫ Σ

dAg¯ (δ K )|Σ +

1 6

∫ Σ

dAg¯ δ s| δn K .

Using Eq. (3.10), and observing that only the variations of IIab contribute when the preinvariant K is written in terms of IIab and γ ab , the first of the terms above becomes

−

1 6

∫

1

Σ

dAg¯ (δ K )|Σ = −

1 3

∫ Σ

¯ a ∇¯ b II˚ ab . dAg¯ δ s| ∇

The second term requires that we compute a normal derivative of K : ab

1

ab

δn K − 2HK = ∇n K = 2II˚ ∇n II˚ ab = 2II˚ [∇n , ∇a ]nb ] ab [ 2 = 2II˚ −Pab − II˚ ab − 2H II˚ ab = −2 tr(II˚ P) − 4HK ,

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M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193 3

where we have used Eq. (2.14) as well as that the Weyl tensor vanishes for d < 4 and for surfaces tr II˚ necessarily vanishes. Assembling the above we find

δ I2 = −

∫

1 3

¯ a ∇¯ b + P⊤ ˚ ˚ ab dAg¯ δˆ s| ∇ ab + H IIab II .

)

(

Σ

Thus, using Eq. (2.12), we obtain the following: Proposition 3.16. With respect to hypersurface variations the gradient of the Willmore energy I2 is 1

− Lab II˚ ab . 3

Thus the Willmore equation is Lab II˚ ab = 0. Note that the Willmore invariant displayed in the above Proposition is manifestly conformally invariant because Lab is the invariant operator of Proposition 2.3. For Euclidean host spaces, Eq. (2.5) says ¯ ∇. ¯ II˚ = ∆ ¯ H while the third of the Gauß Eqs. (2.3) gives tr II˚ 2 = 2H 2 − Sc. Hence using the definition of the operator ∇. ab L in Eq. (2.12), the Willmore equation takes the familiar form

¯ H + 2H(H 2 − K) = 0 , ∆ where K = 21 Sc is the Gauß curvature. A conformally invariant derivation of the Willmore invariant for surfaces into S 3 with its standard conformal structure was given in [7] using a moving frame method. 4. The conformal four-manifold hypersurface energy functional Our main variational result, as stated in Proposition 1.2, is that the obstruction density B3 for spaces agrees with the variational gradient of the rigid membrane functional I3 =

1

∫

6

Σ

dAg¯ L ,

where the membrane rigidity density L is given in Eq. (2.8). In this section we prove Proposition 1.2. We begin by using Lemma 3.6 to write the variation as

δ I3 =

1

∫

6

dAg¯ δ L ⏐Σ −

( )⏐

Σ

1 6

∫ Σ

dAg¯ δˆ s| δnˆ L .

To compute the first of these terms, we use the Fialkow Equation (2.7) to write 3

L = tr II˚ − W (nˆ , II˚ , nˆ ) , so that

∫

dAg¯ δ L ⏐Σ =

( )⏐

Σ

∫

2

[

Σ

ab

dAg¯ 3II˚ ab δ II˚

]⏐ ab n ab n − (δ II˚ )W ab − 2II˚ W bac δ nc ⏐Σ .

Using Eqs. (3.8) and (3.10), this becomes

∫

1

Σ

dAg¯ δ L = 1

∫

[

∫Σ

= Σ

n

n

⏐ ¯ a ∇¯ b II˚ 2(ab)◦ − ∇¯ a ∇¯ b W ab + 2∇¯ c (II˚ ab W ⊤ dAg¯ δ s| 3∇ bac ) Σ n

n

n

]⏐

n

ab bac ⏐ ¯ a ∇¯ b II˚ 2(ab)◦ − ∇¯ a ∇¯ b W ab + W ⊤ + 2II˚ ∇¯ c W ⊤ dAg¯ δ s| 3∇ bac Σ , bac W

[

]⏐

where the final two terms in the last line were obtained using Eq. (2.10) to rewrite the final term of the first line. To compute the second term in δ I3 , we employ Lemma 3.15 to first calculate the normal derivative of the second fundamental form: n

1

⊤ ⊤ 2 (∇n II˚ ab )⊤ = (∇n IIab )⊤ ◦ = ([∇n , ∇a ]nb )◦ = W ab − P(ab)◦ − II(ab)◦ n

2 = 2W ab − P¯ (ab)◦ − 2II˚ (ab)◦ − H II˚ ab ,

where in the final step we have used the Fialkow Equation (2.7). Additionally, we must compute a normal derivative of the Weyl tensor: n

n

1

n

∇n W ab = nd nc ∇c W bad = (g dc − g¯ dc )∇c W bad n

= IIdc Wdabc + nc ∇ d Wdabc − ∇c⊤ W ba c dc

n

n

c n

˚ = II˚ Wdabc − 3H W ab + nc Cbca − ∇¯ c W ⊤ bac + IIa Wcb .

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

181

Orchestrating the above results we find 2

1

ab

ab

n

n

ab

δn L :=∇n L + 3HL = 3II˚ ab ∇n II˚ − (∇n II˚ )W ab − II˚ ∇n W ab n

2 ¯ c II˚ ab W ⊤ ˚ −∇ = 3P¯ ab II˚ (ab)◦ − W (n, P¯ , n) − HW (n, II˚ , n) − C (n, II) bac 2

n

ab cd

n

− 7W (n, II˚ , n) + K 2 + II˚ II˚ Wdabc + 2Wab W ab . Using these, the variation δ I3 becomes 1

δ I3 =

1

∫

6

[

Σ

2

2

n

¯ a ∇¯ b II˚ (ab)◦ + 3P¯ ab II˚ (ab)◦ − ∇¯ a ∇¯ b W ab − W (n, P¯ , n) dAg¯ δ s| 3∇ n

n

n

ab bac ⊤ ˚ ˚ + II˚ ∇¯ c W ⊤ bac − C (n, II) − HW (n, II, n) + W bac W ]⏐ n n 2 ab cd ⏐ − 7W (n, II˚ , n) + K 2 + II˚ II˚ Wdabc + 2W ab W ab ⏐ .

Σ

This result can be written in a manifestly conformally invariant way using the operator Lab of Eq. (2.12) and the hypersurface Bach tensor of Lemma 2.1:

δ I3 =

1 6

∫

(

Σ

nˆ

nˆ

nˆ

2 ab bac dAg¯ δˆ s| Lab 3II˚ (ab)◦ − W ab − II˚ Bab + K 2 + W ⊤ bac W

[

]

) nˆ nˆ 2 ab cd − 7W (nˆ , II˚ , nˆ ) + II˚ II˚ Wdabc + 2W ab W ab . Employing Eqs. (2.7) and (2.9), remembering that the Weyl tensor vanishes identically in three dimensions, we see that the tensor

] nˆ nˆ ) 1 [ ab( 2 2 ab L 2II˚ (ab)◦ + F(ab)◦ − II˚ Bab + 12 K 2 + 2F (ab)◦ Fab + F (ab)◦ II˚ ab + W abc W⊤ abc 6 is the variational gradient which completes the proof of Proposition 1.2. B3 =

5. Conformal hypersurfaces and the singular Yamabe problem We define here a singular Yamabe problem as the boundary version of the classical Yamabe problem of finding a conformally related metric with constant scalar curvature [2,4,31]: Problem 5.1. Given a d ≥ 3-dimensional Riemannian manifold (M , g) with boundary Σ := ∂ M, find a smooth real-valued function u on M satisfying the conditions: (1) u is a defining function for Σ (i.e., Σ is the zero set of u, and dux ̸ = 0 ∀x ∈ Σ ); o (2) g o := u−2 g has scalar curvature Scg = −d(d − 1). Where 0 < u := ρ −2/(d−2) , part (2) of this problem is governed by the Yamabe equation

] [ d−1 d+2 ∆g + Scg ρ + d (d − 1) ρ d−2 = 0 . −4 d−2 But since u is a defining function, to deal with boundary aspects the equation can be usefully recast: Since each g ∈ c is in 1:1 correspondence with a true scale τ ∈ Γ (E M [1]), setting the smooth defining function u = σ /τ , the Yamabe equation becomes ) ( )2 2 ( Sc S(σ ) := ∇σ − σ ∆ + σ = 1. (5.1) d 2(d − 1) In the above ∆ := g ab ∇a ∇b and all index contractions are performed with the conformal metric g and its inverse. It follows go

Sc easily that the quantity S(σ ) on the left hand side of Eq. (5.1) is conformally invariant. Note that S(σ ) = − d(d . Also, −1) because u is a defining function, σ is a defining density: this is a section of Γ (E M [1]) with zero locus Z (σ ) = Σ and such that for any Levi-Civita connection in the conformal class ∇σ ̸ = 0 along Σ ; see Section 5.3. Development of the asymptotic analysis of the singular Yamabe problem as well as uncovering its fundamental role in the theory of conformal hypersurface invariants uses the tractor calculus treatment of conformal geometries. We next review key aspects of that theory.

5.1. Conformal geometry and tractor calculus Although there is no distinguished connection on the tangent bundle TM for a d-dimensional conformal manifold (M , c), there is a canonical metric h and linear connection ∇ T (preserving h) on a related higher rank vector bundle known as the

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tractor bundle T M. This is a rank d + 2 vector bundle equipped with a canonical tractor connection ∇ T [5,39]. The bundle T M is not irreducible but has a composition series given by the semi-direct sum T A M = E M [1] ⨭ Ea M [1] ⨭ E M [−1] .

There is a canonical bundle inclusion E M [−1] → T A M, with T ∗ M [1] a subbundle of the quotient by this, as well as a surjective bundle map T M → E M [1]. We denote by X A the canonical section of T A M [1] := T A M ⊗ E M [1] giving the first of these: X A : E M [−1] → T A M ,

(5.2)

and term X the canonical tractor. A choice of metric g ∈ c determines an isomorphism (or splitting) g

TM ∼ = E M [1] ⊕ T ∗M [1] ⊕ E M [−1].

(5.3)

g

We may write, for example, U = (σ , µa , ρ ), or alternatively [U A ]g = (σ , µa , ρ ), to mean that U is an invariant section of T M and (σ , µa , ρ ) is its image under the isomorphism (5.3); often the dependence on g ∈ c is suppressed when this is clear by context. Changing to a conformally related metric ˆ g = Ω 2 g gives a different isomorphism, which is related to the previous one by the transformation formula (σ , µa , ρ )Ω

2g

( )g = Ω σ , Ω (µa + σ Υa ), Ω −1 (ρ − g bc Υb µc − 21 σ g bc Υb Υc ) ,

where Υb is the one-form Ω −1 d Ω . In terms of the above splitting, the tractor connection is given by

( ∇aT

σ µb ρ

)

( :=

∇a σ − µa ∇a µb + gab ρ + Pab σ ∇a ρ − Pac µc

) .

(5.4)

We will often recycle the Levi-Civita notation to also denote the tractor connection; it is the only connection we shall use on T M, its dual, and tensor powers. Its curvature is the tractor-endomorphism valued two form R♯ ; in the above splitting this acts as ♯

(

Rab

σ µc ρ

)

σ µc ρ

( = [∇a , ∇b ]

)

⎛

0

⎞

= ⎝ Wabc d µd + Cabc σ ⎠ . −Cabc µc

(5.5)

For [U A ] = (σ , µa , ρ ) and [V A ] = (τ , ν a , κ ), the conformally invariant, signature (d + 1, 1), tractor metric h on T M given by h(U , V ) = hAB U A V B = σ κ + gab µa ν b + ρτ =: U · V , T

(5.6) B

is preserved by the tractor connection, i.e., ∇ h = 0. It follows from this formula that XA = hAB X provides the surjection by contraction ιX : T M → E [1]. The tractor metric hAB and its inverse hAB are used to identify T M with its dual and to raise and lower tractor indices. The notation V 2 is shorthand for VA V A = h(V , V ). Tensor powers of the standard tractor bundle T M, and tensor products thereof, are vector bundles that are also termed tractor bundles. We shall denote a tractor bundle of arbitrary tensor type by T Φ M and write T Φ M [w] to mean T Φ M ⊗ E M [w]; w is then said to be the weight of T Φ M [w]. Closely linked to ∇ T is an important second order conformally invariant differential operator ′

DA : Γ (T Φ M [w]) → Γ (T Φ M [w − 1]) , ′

known as the Thomas (or tractor) D-operator. Here T Φ M [w − 1] := T A ⊗ T Φ M [w − 1]. In a scale g,

( [DA ]g =

(d + 2w − 2) w (d + 2w − 2)∇a −(∆g + Jw)

) ,

(5.7)

where here ∆ = g ab ∇a ∇b , and ∇ is the coupled Levi-Civita-tractor connection [5,39]. The following variant of the Thomas D-operator is also useful. Definition 5.2. Suppose that w ̸ = 1 − 2d . The operator

( ′ ) ˆ DA : Γ (T Φ M [w]) −→ Γ T Φ M [w − 1] is defined by

ˆ DA T :=

1 d + 2w − 2

DA T .

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

Remark 5.3. We term the critical value w = 1 −

d 2

183

the Yamabe weight, since one then has

DA = −X A □Y , where the operator −□Y : Γ (T Φ M [1 − 2d ]) → Γ (T Φ M [−1 − 2 J in a choice of scale. ⋆ operator given by ∆ − d− 2

d 2

]) is the conformally invariant, tractor-coupled, Yamabe

The Thomas D-operator is null, in the sense that DA ◦ DA = 0 ; this operator identity can easily be derived from Eq. (5.7). The Thomas D-operator is not, however, a derivation. Its failure to obey a Leibniz rule is captured by the following Proposition, a result which was first observed in [26] and proved in [15]: Proposition 5.4. Let Ti ∈ Γ (T Φ M [wi ]) for i = 1,2, and hi := d + 2wi , h12 := d + 2w1 + 2w2 − 2 with hi ̸ = 0 ̸ = h12 . Then

ˆ DA (T1 T2 ) − (ˆ DA T1 ) T2 − T1 (ˆ DA T2 ) = −

2 d + 2w1 + 2w2 − 2

X A (ˆ DB T1 )(ˆ DB T2 ) .

(5.8)

We shall also need the following Lemma which is easily verified by direct application of Eq. (5.7) or the modified Leibniz rule (5.8) in tandem with the identity

ˆ DA X B = hAB .

(5.9)

Lemma 5.5. Let T ∈ Γ (T Φ M [w]). Then DA (X A T ) = (d + w )(d + 2w + 2)T .

(5.10)

5.2. Conformal hypersurfaces and tractors The basis for a conformal hypersurface calculus [12,15,17,24,38,40] is a unit tractor object N A ∈ Γ (T M)|Σ (from [5]) corresponding to the conformal unit conormal nˆ , termed the normal tractor. This is defined along Σ , in a choice of scale, by

( A

[N ] =

0 nˆ a −H

) ⇒ NA N A = 1 .

(5.11)

The subbundle T M ⊤ orthogonal to the normal tractor (with respect to the tractor metric h) along Σ is canonically isomorphic to the intrinsic hypersurface tractor bundle T Σ , see [6] and [12, Section 4.1]. This is the conformal tractor analog of the Riemannian isomorphism between the intrinsic tangent bundle T Σ and the subbundle TM ⊤ of TM |Σ orthogonal to nˆ a along Σ . This isomorphism identifies these bundles and thus we use the same abstract index for T M and T Σ . In explicit computations in a given choice of scale, we will need to relate sections given in corresponding splittings of the ambient and hypersurface tractor bundles. In terms of sections expressed in a scale g ∈ c (which determines g¯ ∈ c¯ ), the isomorphism is given by

v+ [ A] V g := va v−

(

⎛

)

⎞

v+

]g [ ] [ ] ⎟ [ ⎜ ↦→ ⎝ va − nˆ a H v + ⎠ = U A B g¯ V B g =: V¯ A g¯ , v + −

(5.12)

v

1 2 + H 2

where V A ∈ Γ (T M ⊤ ), V¯ A ∈ Γ (T Σ ) and the SO(d + 1, 1)-valued matrix

⎛ U AB

[

]g g¯

1

⎜ := ⎝ −ˆna H − 12 H 2

⎞

0

0

δab

0⎠ .

nˆ b H

⎟

1

In the above the canonical isomorphism between T ∗ M ⊤ ⏐Σ and T ∗ Σ defined by the unit normal vector nˆ is used to identify sections of these bundles. The ambient tractor connection ∇ obeys an analog of the classical Gauß formula relating it to the intrinsic tractor ¯ of (Σ , c) which is known as the Fialkow–Gauß formula [24,38,40] or in our current notation connection ∇ [17, Proposition 4.14] because the Fialkow tensor of Eq. (2.7) encodes the difference between these two connections along Σ . As in the Gauß case, the tractor-coupled operator ∇ ⊤ := ∇ − nˆ ∇nˆ is well-defined along Σ irrespective of how tensors are extended to the ambient manifold. For future use, we record some identities for this operator acting on the normal tractor:

⏐

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Lemma 5.6. Along Σ , tangential, tractor-coupled gradients of the normal tractor are given in a choice of scale by

⎛

0

⎜ ⎟ ⎟ ˚ ∇a N = ⎜ ⎝ IIac ⎠ , ¯ ˚ IIa − ∇. d¯ −1 ⊤

⎛

⎞

C

⎞ −II˚ ab

⎜ ⎜ 1 ∇a ∇b N = ⎜ ¯ II˚ b ⎜ ∇a⊤ II˚ bc − d¯ −1 g¯ac ∇. ⎝ ¯ II˚ b − II˚ cb P⊤ − d¯ −1 1 ∇a⊤ ∇. ac ⊤

⊤

C

⎟ ⎟ ⎟, ⎟ ⎠

(5.13)

while the tangential, tractor-coupled operator ∆⊤ := g ab ∇a⊤ ∇b⊤ gives

⎛

0

⎞

⎟ ⎜ ¯ ⎜ d−2 ¯ II˚ c − nc K ⎟ ∆⊤ N C = ⎜ d¯ −1 ∇. ⎟. ⎠ ⎝ ¯ ∇. ¯ II˚ +(d¯ −1)Pab II˚ ab ∇. − ¯d−1

(5.14)

Proof. Since ∇ ⊤ is tangential, the first identity follows directly from [17, Corollary 3.9]. The third identity follows directly from the second, both of which only require a direct application of Eq. (5.4). □ Remark 5.7. Note that expressions such as ∇a⊤ II˚ bc in the above Lemma are well-defined along Σ since ∇ ⊤ is tangential. It is obviously possible to further develop the second and third identities using the Gauß formula (2.2) and (for d¯ ≥ 3) the Fialkow–Gauß Equation (2.7). ⋆ 5.3. Singular Yamabe problem asymptotics and defining densities Given a hypersurface Σ , a section σ ∈ Γ (E [1]) is said to be a defining density for Σ if Σ = Z (σ ) and ∇σ is nowhere vanishing along Σ , where ∇ is the Levi-Civita connection for some g ∈ c. The conformal analog of the exterior derivative of a defining function is then provided by the tractor field 1 g IσA := ˆ DA σ = (σ , ∇a σ , − (∆ + J)σ ) =: (σ , na , ρ ) . d In Riemannian signature for any defining density σ , we have that

(5.15)

Iσ2 > 0 holds in a neighborhood of Σ . More generally, any section σ ∈ Γ (E M [1]) such that IσA is nowhere vanishing is called a scale, IσA is its scale tractor and the data (M , c , Σ ) is an almost Riemannian structure (or geometry). The following proposition, whose proof follows directly from Eqs. (5.15) and (5.6), shows that almost Riemannian geometries are a natural setting for the singular Yamabe problem: Proposition 5.8 (see [12]). For σ ∈ Γ (E [1]) the quantity S(σ ) of Eq. (5.1) is the squared length of the corresponding scale tractor: S(σ ) = Iσ2 := hAB IσA IσB .

(5.16)

Indeed, the Yamabe equation (5.1) now reads simply Iσ2 = 1 . Thus, the asymptotic version of the singular Yamabe Problem 5.1 is stated as follows: Problem 5.9. Find a smooth defining density σ¯ such that Iσ2¯ = 1 + σ¯ ℓ Aℓ ,

(5.17)

for some smooth Aℓ ∈ Γ (E [−ℓ]), where ℓ ∈ N ∪ ∞ is as high as possible. The above problem was solved in [15,16], that result is captured by the following Theorem whose proof may be found in [15]: Theorem 5.10. Let σ be a defining density for a hypersurface Σ and σ0 := σ / Iσ2 , then there exist smooth densities Ak ∈ Γ (E Σ [−k]) such that

√

( ) σ¯ = σ0 1 + A1 σ0 + A2 σ02 + · · · + Ad σ0d solves Iσ2¯ = 1 + σ0d Bσ ,

(5.18)

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

185

for some smooth Bσ ∈ Γ (E M [−d]). For 1 ≤ k ≤ d the densities Ak are determined by the recursion

[ σ¯ k = σ¯ k−1 1 −

]

Iσ2¯k−1 − 1

d

2 (d − k)(k + 1)

where σ¯ k−1 := σ0 (1 + τ ∈ Γ (E M [1]),

∑k−1 j=1

,

σ j Aj ) solves Iσ2¯k−1 = 1 + σ k−1 C for smooth C ∈ Γ (E M [1 − k]). Moreover, for any true scale

[ ] I2 − 1 d σ ′ = σ¯ 1 + log(σ¯ /τ ) σ¯ 2 d+1 solves Iσ2 ′ = 1 + σ d+1 C ′ + D log(σ /τ ) + D′ log2 (σ /τ )

(

)

for smooth C ′ , D, D′ ∈ Γ (E M [−d − 1]). Finally, given two defining functions σ and σ˜ for Σ , then Bσ ⏐Σ = Bσ˜ ⏐Σ =: Bd .

⏐

⏐

The last statement of this theorem implies that the density Bd is uniquely determined by (M , c , Σ ). Since it obstructs a smooth all orders solution to the singular Yamabe problem we term it the obstruction density. The solution σ¯ (σ ) is determined uniquely by the recursion up to the order of the obstruction. Thus, up to the given order, σ¯ is a canonical scale defining the hypersurface Σ and whose scale tractor has unit length to accuracy σ d , so is termed a conformal unit defining density. 5.4. Conformal hypersurface invariants and canonical extensions A Riemannian hypersurface invariant P (Σ ; g) := P (s; g)|Σ with the property P (Ω s, Ω 2 g) = Ω w P (s; g) is termed a conformal hypersurface covariant. This determines an invariant section P (σ , g ) ∈ Γ (E M [w]) where σ is a defining density. Thus P (σ , g )|Σ is called a conformal hypersurface invariant. As for the Riemannian case, this definition extends to tensor and therefore also tractor-valued invariants. Since the conformal unit scale σ¯ is a unique defining density, up to the addition of terms σ¯ d+1 S for S ∈ Γ (E M [−d]), it follows that any conformal density P (σ¯ (σ ), g ), where σ¯ (σ ) is determined in terms a given defining density σ via Eq. (5.18) of Theorem 5.10, gives a conformal hypersurface invariant P (σ¯ (σ ), g )|Σ so long as the formula for P involves jets of σ¯ of order ≤ d. (In fact, P (σ¯ (σ ), g ) is a density-valued preinvariant.) This construction provides holographic formulae for conformal hypersurface invariants, see [15, Section 6]. Example 5.11. The scale tractor for the conformal unit defining density gives a holographic formula for the normal tractor IσA¯ |Σ = ˆ DA σ¯ |Σ = N A . This result was first proved in [12]. ⋆ Moreover, in the above example we may view the conformal unit defining density as providing a canonical extension IσA¯ of the normal tractor N A . A short computation (in a choice of scale) shows that

(

0

)

∇a Iσ¯ = ∇a nb + Pab σ¯ + gab ρ ⋆ B

,

where ρ := − 1d (∆σ¯ + Jσ¯ ) as in Eq. (5.15). Comparing this with Eq. (5.13) gives a holographic formula for the trace-free second fundamental form II˚ ab = (∇a nb + Pab σ¯ + gab ρ )|Σ . Hence we have the following canonical extensions of the trace-free second fundamental form and rigidity density II˚ ab := ∇a nb + Pab σ¯ + gab ρ and K := (∇a nb + Pab σ¯ + gab ρ )(∇ a nb + Pab σ¯ + g ab ρ ) .

(5.19)

The above formulæ are now well-defined in M rather than only along Σ . These extensions are canonical up to a finite order determined by Theorem 5.10. 5.5. The Laplace–Robin operator and tangential operators By contracting the scale tractor and Thomas D-operator, we can build a new operator that plays two roles: that of the ambient Laplace operator and of a conformally invariant boundary Robin-type Dirichlet-plus-Neumann operator [11]. Thus we call the operator I · D : Γ (T Φ M [w]) −→ Γ (T Φ M [w − 1]) ,

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the Laplace–Robin operator. Calculated in some scale g ∈ c

w

2w (d + w − 1)σ J , (5.20) d d where n = ∇σ . Hence this is a Laplace-type operator which is degenerate along Σ = Z (σ ). Indeed, along Σ the Laplace– Robin operator becomes first order. In particular, for a conformal unit defining scale, along Σ and in a choice of scale we have I · D = −σ ∆ + (d + 2w − 2) ∇n −

[

Σ

(∆σ ) −

]

Iσ¯ · D = (d + 2w − 2) ∇n − w H .

(

)

Hence, for w ̸ = 1 − 2d and in the spirit of Section 5.4, along Σ the Laplace–Robin operator is a operator-valued holographic formula for the Robin operator δnˆ as given in Eq. (2.14) (recall that for a conformal unit defining density, along Σ one has ∇σ = nˆ ). Given a natural density, tensor or tractor field f¯ along Σ , it is often the case that we can fix an extension f off Σ , that is canonical to some order using the conformal unit defining density, such as those given in Eq. (5.19)). In that case we will denote the Robin operator by

{ δR f¯ =

Iσ¯ · ˆ Df ⏐Σ ,

⏐

⏐ ∇nˆ f ⏐Σ +

d−2 2

H f¯ ,

w ̸= 1 − 2d , w = 1 − 2d .

In light of this, we will often use the same notation for f and f¯ . Combining the conformal unit defining density and the Laplace–Robin operator allows successive normal derivatives of canonically extended conformal hypersurface invariants to be defined up to the order of the obstruction. The canonical extensions of the normal tractor and intrinsic canonical tractor are the scale tractor IσA¯ and the ambient canonical tractor X A , respectively. The following Lemma gives one normal derivative acting on these: Lemma 5.12. The Robin operator acting on the canonically extended canonical and normal tractors gives:

δR X A = N A ,

δR N A =

KX A d¯ − 1

.

(5.21)

Proof. The first identity is a direct consequence of Eq. (5.9). For the second we compute in a scale

( ⏐ δR N = ∇n Iσ¯ ⏐Σ = A

A

0

)

∇n na − Hna ∇n ρ − P(n, n)

.

To complete the proof we need the normal derivatives of the canonical extensions of na and ρ determined by IσA¯ . These were computed in [17, Lemmas 3.3 and 3.8] and, along Σ are:

∇n na = Hna ,

∇n ρ = P(n, n) +

K d¯ − 1

. □

(5.22)

The Laplace–Robin operator has one further crucial role to play. In [14], it was shown that by viewing the scale σ as an operator mapping Γ (T Φ M [w]) → Γ (T Φ M [w + 1]) and acting by multiplication, then the commutator of this with the Laplace–Robin operator acting on weight w tractors obeys

[Iσ · D, σ ] = −Iσ2 (d + 2w) . This identity underlies a solution-generating sl(2) algebra. A particular consequence of this is that the operator

(

Pkσ := −

1 Iσ2

)k

Iσ · D

is tangential when acting on weight w = differential operator Pk : Γ T Φ M

(

k−d+1 2

tractors. Specializing to the conformal unit defining density gives a

[ k − d + 1 ])⏐⏐ ( [ −k − d + 1 ])⏐⏐ ⏐ −→ Γ T Φ M ⏐ ,

Σ Σ 2 2 which is completely determined by the data (M , c , Σ ) [15,17]. We call this the extrinsic conformal Laplace operator. For k k even the operator Pk is (∆⊤ ) 2 + L.O.T., up to multiplication by a non-vanishing constant, where ‘‘L.O.T.’’ stands for terms of lower derivative order involving both intrinsic and extrinsic curvature quantities.

6. Singular Yamabe obstruction densities Given a unit conformal defining density, the obstruction density Bd¯ has a simple holographic formula whose leading structure is given in terms of the extrinsic conformal Laplace operator [15, Theorem 7.7]:

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

187

Theorem 6.1. The obstruction density is given by Bd¯ =

2

(

[

]⏐ )]

¯ A ΣBA Pd¯ N B + (−1)d¯ −1 I¯ · Dd¯ −1 (X B K ) ⏐ D Σ

d¯ !(d¯ + 1)!

[

,

(6.1)

where the rigidity density K is canonically extended off Σ by the formula K = PAB P AB with P AB := ˆ DA I¯B . Also, the projector ΣBA := δBA − N A NB . The main aim of this section is to compute B3 using the above holographic formula. The canonical extension of the rigidity density K stated in the theorem is the same as that given in Eq. (5.19). Also, in hypersurface dimensions d¯ = 2, 3, the extrinsic conformal Laplacians acting on weight zero tractors are given by P2 = ∆⊤ ,

(

)

ab ¯ II˚ b ∇b⊤ + 4na R♯ ◦ ∇ b + ∇ b ◦ R♯ . P3 = −8 II˚ ∇a⊤ ∇b⊤ − 8 ∇. ab ab

(6.2)

Proposition 6.2. If d¯ = 2, then ab

P2 N C = −X C Lab II˚

− NC K ,

d¯ = 2 .

Proof. Using Eqs. (6.2) and (5.14), we have for d¯ = 2,

⎛

⎞

0 ⎠. − nc K P2 N C = ∆⊤ N C = ⎝ ¯ ∇. ¯ II˚ − Pab II˚ ab −∇. Using Eq. (2.12) the quoted result follows. □ Remark 6.3. Our computation of B3 relies on a d¯ = 3 version of the above Proposition. ⋆ The second term in Eq. (6.1) involves normal derivatives of the canonically extended rigidity density K . The following proposition provides the first of these. Proposition 6.4. density L obey

˚ rigidity density K and membrane rigidity The canonically extended trace-free second fundamental form II,

nˆ ( )⊤ 1 2 δR II˚ ab = −II˚ ab + W ab + g¯ab K ,

2

δR K = −2(d¯ − 2)L .

(6.3)

Proof. We begin by computing a normal derivative of the canonical extension of II˚ ab :

∇n II˚ ab =∇n ∇a nb + gab ∇n ρ + σ ∇n Pab + Pab ∇n σ n ( ) =Rab − (∇a nc )∇ c nb − ∇a ∇b (ρσ ) + gab ∇n ρ + Pab + σ ∇n − 2ρ Pab n

2

=Rab − II˚ ab + ρ II˚ ab − 2n(a ∇b) ρ + gab ∇n ρ + Pab [( ] ) + σ ∇n − 3ρ Pab + 2II˚ c (a Pcb) − ∇a ∇b ρ + O(σ 2 ) .

(6.4)

Projecting onto the tangential piece of this quantity along Σ , and using Eq. (5.22) we find

(

∇n II˚ ab

)⊤

Σ

2

n

1

= −II˚ ab + W ab + g¯ab K − H II˚ ab . 2

Using Eq. (2.15), we have n ]⊤ Σ ( )⊤ Σ [ 1 2 δR II˚ ab = (∇n + H)II˚ ab = −II˚ ab + W ab + g¯ab K .

2

ab The second equation follows by noting that δR K = 2II˚ δR II˚ ab and using the above result. □

Remark 6.5. Recalling that the Willmore energy functional (3.11) for embedded surfaces is the integral of the rigidity density K and its analog (1.1) for embedded spaces is the integral of the membrane rigidity density L, we see from Eq. (6.3) that the Robin operator relates the two integrands for these functionals. It would be interesting to investigate whether this phenomenon holds in higher dimensions d¯ = 2k to d¯ = 2k + 1 (k ∈ Z≥2 ). ⋆ Using Propositions 6.2 and 6.4, as well as Eqs. (5.21) and (5.10), we arrive at an explicit formula for the obstruction density for surfaces.

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Proposition 6.6. The obstruction density for d¯ = 2 is given by 1

Lab II˚ ab . 3 Comparing the above with Proposition 3.16 verifies again that the obstruction density and the gradient of the Willmore energy I2 agree. B2 = −

6.1. Four-manifold obstruction density Our goal is now to find hypersurface formula for the d¯ = 3 obstruction density and thus prove Proposition 1.1. We begin by calculating P3 N B . Proposition 6.7. When d¯ = 3, P3 N B = 4 B′ X B +

( ) B 4 [ −1 ]B C ¯ U C D K − 4 δR K N , 3

where nˆ

2

ab

B′ =Lab 2II˚ (ab)◦− W ab − II˚ Bab +

(

)

1 2

nˆ

2

nˆ

K 2 − 3W (nˆ , II˚ , nˆ ) + W ab W ab +

1 2

nˆ

nˆ

abc W⊤ . abc W

(6.5)

Proof. Using Eq. (5.13), the first two terms in P3 N C appearing in Eq. (6.2) can be written explicitly as 8K ( 3 ) ¯ ⎝ −8II˚ ∇¯ a II˚ bc − ∇.II˚ b + 8nc tr II˚ + HK ⎠ . ab ¯ II˚ a + 8 tr(II˚ 2 P⊤ ) ¯ II˚ b + 4∇. ¯ II˚ a ∇. ¯ a ∇. 4II˚ ∇

⎞

⎛

ab

b 4II˚ c

To compute the remaining two terms of P3 N C , we make use of the tractor curvature and connection formulæ in Eqs. (5.5) and (5.4), respectively, as well as Eq. (5.13), and find

⎛ Σ

♯

na ∇ b ◦ Rab N C = ⎝

⎞

0 n

ab

−W abc II˚

⎜ n

−∇¯ b C b − W (n, P ⊤ , n) ⎛ ⎞

⎟ ⎠,

0 n Σ ♯ na Rab ◦ ∇ b ⊤ N C = ⎝−W abc II˚ ab ⎠ . ˚ −C (n, II)

In the first equation we have made use of the four-manifold identity ∇ b Wbacd = Ccda . Orchestrating, we have

⎛

⎞ ) ⎜ ¯ a II˚ bc − ¯ II˚ b − 8W (n, II˚ , c)⊤ + 8nc tr II˚ − W (n, II˚ , n) + HK ⎟ ∇. P3 N C = ⎝ −8II˚ ∇ ⎠. n ab a 2 ⊤ b ⊤ ¯ ¯ ¯ ¯ ¯ ˚ ˚ ˚ ˚ ˚ ˚ 4II ∇a ∇.IIb + 4∇.IIa ∇.II + 8 tr(II P ) − 4C (n, II) − 4∇ C b − 4W (n, P , n) 8K

ab

b 4II˚ c

3

(

This expression can be simplified by comparing it with the boundary Thomas D-operator acting on the rigidity density,

⎛

⎞⎛

⎞

⎛

⎞

8K 8K 0 ¯ c K ⎠ = ⎝ −4∇¯ b K + 8nb HK ⎠ . 0⎠ ⎝ −4∇ ¯ − 2J¯ )K ¯ − 2J¯ )K − 4H 2 K 1 − 34 (∆ − 43 (∆

1 0 4 [ −1 ]B C c ¯ ⎝ n H δ U D K = b C b 3 1 2 c − 2 H −n H

(6.6)

Here we have used the isomorphism (5.12) between sections of T Σ and T M ⊤ . To match this expression with terms appearing in P3 N C as displayed above, we make use of the trace-free Mainardi Equation (2.10) to obtain the identity 1

n 1 b ab ab ¯ II˚ b + W ⊤ ˚ ab ∇¯ c K = II˚ ∇¯ c II˚ ab = II˚ ∇¯ a II˚ bc + II˚ c ∇. abc II .

2 2 Using this identity, Eq. (2.13), the Fialkow Equation (2.7) as well as Eq. (6.3) we have P3 N B =

4[ 3

U −1

]B

¯ C K − 4N B δR K + 4B′ X B ,

CD

with B′ given by B′ =

2 3 1

nˆ 4 8 bc ¯ II˚ a )(∇. ¯ II˚ a ) + 4 II˚ ab ∆ ˚2 ¯ II˚ ab − J¯ K − II˚ ab ∇¯ c W ⊤ ∇¯ a II˚ ∇¯ a II˚ bc + (∇. abc − 3W (n, II , n)

3

2

n

n

+ K + W ab W 2

ab

n

3

n

3

ab ¯ ¯b ˚ ˚ + 2II˚ ∇¯ c W ⊤ bac − ∇ C b − W (n, P , n) − C (n, II) − HW (n, II, n) .

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

189

It remains only to show that this last expression reduces to that stated in the proposition. This follows directly from Proposition 2.4 and the identities n

n

n

ac

¯ c W cb + W ⊤ ˚ Cb = ∇ acb II

n n n 1 n ⊤ n abc ab W abc W ⇒ ∇¯ b C b = ∇¯ a ∇¯ b W ab + II˚ ∇¯ c W ⊤ ; abc +

2

these are direct consequences of Eqs. (1.2) and (2.10). Noting that n = nˆ along Σ completes the proof. □ We now turn to the second term in the holographic formula (6.1). We first need the following technical result: Lemma 6.8. Let T ∈ Γ (T φ M [−2]) and d¯ = 3. Then Σ

I · D2 (X C T ) = −2DC T . Proof. The key to the proof is to note that for d = 4, the tractor X C T has ambient Yamabe weight w = −1, so that I · D(X B T ) = −σ □Y (X B T ) . Σ

Acting with the remaining I · D = −2δR (at this weight) and calculating explicitly using the tractor connection (5.4) gives the identity stated. □ Our result for two canonical normal derivatives of the rigidity density multiplied by the canonical tractor is given by the following: Proposition 6.9. Let d¯ = 3. Then, Σ

I · D2 (X C K ) = 4X B B′′ −

4 [ −1 ]B C B ¯ U C D K + 4N δR K , 3

(6.7)

where 2 B′′ = Lab II˚ (ab)◦ +

1 2

nˆ

nˆ

2 ab cd K 2 − 4W (nˆ , II˚ , nˆ ) + W ab W ab + II˚ II˚ Wcabd +

1 2

nˆ

nˆ

abc W⊤ . abc W

(6.8)

Proof. Using Lemma 6.8 we must compare

−8K 4∇c K −2D K = 2(∆ − 2J)K

(

)

C

with Eq. (6.6). This yields Eq. (6.7) with B′′ =

1 2

1

3

1

1

6

2

3

4

¯ K − H 2 K − J¯ K − K 2 − P(n, n)K . ∇n2 K + 2H δR K + ∆

(6.9)

It remains to show that this expression reduces to Eq. (6.8). For that, we need to compute two normal derivatives of the canonically extended rigidity density K . Returning to Eq. (6.4), we see that the following normal derivatives of curvature terms appear in 12 ∇n2 K : n

∇n Rab + 2∇n Pab = (∇n nd )Rdabe ne + nc nd ∇c (Rdabe ne ) + 2∇n Pab Σ

n

= H Rab + (g cd − g¯ cd )∇c (Rdabe ne ) + 2∇n Pab n

n

= H Rab + ∇ d (Rdabc nc ) + ∇d⊤ Rb d a + 2∇n Pab n

n

= H Rab + (∇ d nc )Rdabc − nc (∇b Rdac d + ∇c Rda d b ) + ∇d⊤ Rb d a + 2∇n Pab Σ

n

n

dc

= H Rab + (II˚ + Hg dc )Rdabc + nc ∇b Rac − gab ∇n J + ∇d⊤ Rb d a . ab

In the fourth line above we used the Bianchi identity for the Riemann tensor. Upon contraction with II˚ n

ab 2 ˚ + II˚ ab II˚ cd Wcabd II˚ ∇n Rab + 2∇n Pab = W (n, II˚ , n) − HW (n, II˚ , n) − 3H tr(IIP)

(

)

n

n

2 ab ab − tr(II˚ P) + II˚ ∇¯ a P⊤b + K P(n, n) − II˚ ∇¯ d W ⊤ bad .

the above yields

190

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

Combining the above expression and Eq. (6.4), we have n

ab ab ab c Σ II˚ ∇n2 II˚ ab = II˚ ∇n Rab + 2∇n Pab − 2II˚ II˚ cb ∇n II˚ a + K ∇n ρ

(

)

ab

2

ab

˚ ⊤ ) + 2 tr(II˚ P⊤ ) − II˚ ∇a ∇b ρ + ρ II˚ ∇n II˚ ab − 3ρ tr(IIP 1

=

3

ab ¯ II˚ ab − II˚ ∆

1 3

3 J¯ K + 2H tr II˚ +

1 2

H 2K +

1 4

2 K 2 − 2W (n, II˚ , n)

4 ab ab cd − 2HW (n, II˚ , n) + II˚ II˚ Wcabd + K P(n, n) − II˚ ∇¯ c W ⊤ bac . n

(6.10)

3

4 Here we have used Eqs. (5.22), (6.4) and (2.7) as well as the 3 × 3 matrix identity tr II˚ = ab term II˚ ∇a ∇b ρ , the following consequence of the Mainardi Equation (2.4) is needed: ab ¯ a ∇¯ b H = II˚ ∇

1 3

n

ab ¯ II˚ ab − II˚ ab ∇¯ aP⊤ II˚ ∆ b −

The computation of ∇

2 nK

1 3

2

¯ − J¯ K − tr(II˚ P)

3

In addition, to handle the

n

ab c ¯ W⊤ II˚ ∇ bac .

ab also requires the cross term (∇n II˚ ab )(∇n II˚ ). Again using Eq. (6.4) we have

n

Σ

1

1 2 K . 2

2

1

¯ II˚ − H II˚ ab − II˚ ab + gab K . ∇n II˚ ab = W ab − na nb K + n(a ∇. b) 2 Contracting this identity on itself we find n

n

ab 2 (∇n II˚ ab )(∇n II˚ ) = W ab W ab − 2HW (n, II˚ , n) − 2W (n, II˚ , n) + 3

1 2

¯ II˚ b )(∇. ¯ II˚ b ) (∇.

1

+ H 2 K + 2H tr(II˚ ) + K 2 . 2

Orchestrating this and Eq. (6.10), we have 1 2

∇n2 K =

1 2

¯ II˚ a )(∇. ¯ II˚ a ) + (∇.

1 3

ab ¯ II˚ ab + II˚ ∆

3 4

n

n

2 K 2 − 4W (n, II˚ , n) + W ab W ab +

1

4

3 2

H 2K n

3 ab cd ab − 4HW (n, II˚ , n) + 4H tr II˚ − J¯ K + II˚ II˚ Wcabd + P(n, n)K − II˚ ∇¯ c W ⊤ bac .

3 Substituting this result into Eq. (6.9) and simplifying gives B′′ =

1 3

3

n 1 2 4 bc ¯ II˚ a )(∇. ¯ II˚ a ) + 2 II˚ ab ∆ ¯ II˚ ab − J¯ K − II˚ ab ∇¯ c W ⊤ ∇¯ a II˚ ∇¯ a II˚ bc + (∇. bac

2 3 3 3 n n 1 2 ab cd ab ˚ ˚ K + W ab W + II II Wcabd . 2 Using Proposition 2.4 gives the result (6.8), which completes the proof. □ 2 − 4W (n, II˚ , n) +

We now complete the proof of Proposition 1.1 giving the obstruction density for four-manifolds: Proof of Proposition 1.1. Using Eq. (6.1) and Propositions 6.7 and 6.9, the obstruction density is given by B3 =

1

¯ A 4XA (B′ + B′′ ) . D

[

]

72 Substituting the formulæ (6.5) and (6.8) for B′ and B′′ into the above and using Eq. (5.10) gives the result when Eqs. (2.7) and nˆ

⊤ (2.9) are used to trade Wabcd and W ab for the Fialkow tensor and trace-free second form.

6.2. Examples and umbilicity Our results can be easily applied to explicit metrics and hypersurfaces. As we shall explain below, this gives both an independent check of our computations and also generates interesting examples. For surfaces, total umbilicity suffices to ensure vanishing of the obstruction density (this follows by inspection of the formula for B2 given in Proposition 6.6). The following proposition shows this is not true in dimension d¯ = 3. Proposition 6.10. Total umbilicity (i.e., vanishing of II˚ everywhere) is not a sufficient condition for the obstruction density to vanish. Proof. See the explicit counterexample on the third line of the table below. □ Remark 6.11. There is no in principle difficulty constructing examples of umbilic hypersurfaces with non-vanishing obstruction density in dimensions d¯ > 3. Also, albeit in Lorentzian signature, the Schwarzschild example below provides another d¯ = 3 example of this. ⋆

M. Glaros, A.R. Gover, M. Halbasch et al. / Journal of Geometry and Physics 138 (2019) 168–193

191

In the following table we give some sample four manifold metrics, hypersurfaces and their obstruction density B3 . Additionally, σ¯ denotes the conformal unit density in the scale determined by the quoted metric, H the mean curvature, II˚ the trace-free second fundamental form and L the membrane rigidity density. Non-vanishing expressions that are too long to be displayed are denoted by a ⋆. These results were generated with the aid of a computer software package [35].

σ¯

Σ

ds2 dx2 +(1+α x)(dy2 +dz 2 +dw 2 )

x= 0

dx2 +(1+α x)dy2 +(1−α x)dz 2 +dw 2

x= 0

dx2 +(1+z)dy2 +(1+w )dz 2 +(1+y)dw 2 dr 2 +r 2 dΩ 2 2 −(1− 2M r )dt + 1− 2M r

2 3 x+ α x2 − α x3 + 5 α x4 4 8 64 2 x− α x3 8

⋆ ⋆

x= 0 r =R+2M

H

II˚

B3

L

α 2

0

0

0 0 Eq. (6.11)

α (dy2 −dz 2 )

α4

0 Eq. (6.11)

⋆

0 0 0 Eq. (6.12)

2

12

Eq. (6.13)

The first line of the table gives a metric and an umbilic but non-minimal hypersurface for which the obstruction density vanishes. In the second line, the hypersurface is minimal but non-umbilic. The third line is particularly interesting, the hypersurface is both umbilic and minimal, yet the obstruction density does not vanish. Finally in the fourth line, we use the fact that all local formulæ presented in this paper can be extended to Lorentzian structures (provided the conormal is nowhere null), and study a spherical shell of radius R in the Schwarzschild metric. The conormal dr is spacelike for R > 0, timelike for R < 0 and lightlike at the horizon R = 0, so that strictly the formulæ presented below apply only to the spacelike case. In that case the second fundamental form is given by 1

H=

3

M + 2R

√

R(R + 2M)3

,

1 II˚ = − (M − R) 3

√

R + 2M ( R

2Rdt 2 (R +

2M)3

) + dΩ 2 ,

(6.11)

so that the hypersurface Σ is neither minimal nor umbilic, save for the distinguished radius R = M where umbilicity holds. At that value the membrane rigidity density also vanishes; at general R it is given by L=

2(M − R)(M 2 + 6MR + R2 )

√

R R(R + 2M)9

.

(6.12)

The obstruction density is B3 =

2M 4 + 4M 3 R + 30M 2 R2 − 8MR3 − R4 27R2 (R + 2M)6

(6.13)

and has one simple real root in the region R > −2M at R/M ≈ 2.9. Finally let us explain how the formulæ for the obstruction density can be generated without simply computing each term in Proposition 1.1. This is based on the two iterative recursions given in [15, Propositions 2.5 and 4.9]. Firstly, given a four manifold with metric g and hypersurface Σ labeled by a defining function s, we improve the defining function to s s0 = |∇ s|

⏐

which ensures that |∇ s0 |⏐Σ = 1. It is highly propitious to work with a unit defining function s1 with |∇ s1 | = 1. For applications, it suffices to require |∇ s1 | = 1 + sℓ f for smooth f and ℓ of sufficiently high order. This is achieved using the recursion of [15, Proposition 2.5]. For the first three examples tabulated above, s1 = x is already a unit defining function, but for the Schwarzschild case, we performed this algorithm explicitly using the computer software package [35]. Supposing now that s is a unit defining function, we next improve this to a conformal unit defining density evaluated in the scale determined by the metric g. This can be achieved using the recursion of [15, Proposition 4.9]. In practice this simply amounts to repeatedly computing the leading failure of Iσ2 to equal unity and then adding a next to leading correction to σ to correct for this. In the Appendix we present the first four improvement terms for a unit defining function s as well as the corresponding obstruction densities for hypersurface dimensions d¯ = 1, 2, 3 and 4. These were computed using the symbolic manipulation software [27]. We have verified their correctness by checking that the resulting Iσ2 equals unity to the required order for explicit metric and hypersurface examples, including those given above. We note that the result for B4 quoted in the appendix, when evaluated along Σ gives the obstruction density for fivemanifold hypersurfaces expressed in terms of a unit defining function. This is a novel result that is simple to use in practical applications. There is no in principle difficulty employing the unit defining function identities of Sections 3 and 4 and their higher order analogs to evaluate B5 in terms of hypersurface invariants. Acknowledgments A.G. and A.W. would like to thank R. Bonezzi, C.R. Graham and Y. Vyatkin for helpful comments and discussions. The authors gratefully acknowledge support from the Royal Society of New Zealand via Marsden Grant 13-UOA-018 and the UCMEXUS-CONACYT, Mexico grant CN-12-564. A.W. thanks for their warm hospitality the University of Auckland and the Harvard University Center for the Fundamental Laws of Nature. A.W. was also supported by a Simons Foundation, USA Collaboration Grant for Mathematicians ID 317562.

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Appendix. Conformal unit defining function formulæ and five-manifold obstruction density Let s be a unit defining function for a hypersurface Σ in a d-dimensional Riemannian manifold (M , g). The following formulæ improve this to sk which determines a defining density σk in the scale g such that Iσ2k = 1 + σ k f ,

k < d,

for some smooth f ∈ Γ (E M [−k]): s0 = s s1 = s0 + s2 = s1 + s3 = s2 +

1 2(d − 1) 1

s2 ∇.n

6(d − 2)(d − 1) 1

s3 2(∇.n)2 − (d − 4) ∇n ∇.n + 2(d − 1) J

(

24(d − 3)(d − 2)(d − 1)

s4

( 5d − 6 d−1

)

(∇.n)3 − 4(2d − 9) ∇.n ∇n ∇.n

) + (d − 4)(d − 6) ∇n2 ∇.n + 3(d − 2) ∆∇.n + 4(2d − 3) ∇.n J − 2(d − 6)(d − 1) ∇n J ( 13d3 − 62d2 + 100d − 48 1 s4 = s3 + s5 (∇.n)4 120(d − 4)(d − 3)(d − 2)(d − 1) (d − 2)(d − 1)2 398d2 − 880d − 49d3 + 576 13d3 − 156d2 + 524d − 384 + (∇.n)2 ∇n∇.n + ∇.n ∇n2 ∇.n (d − 2)(d − 1) d−1 25d2 − 94d + 72 11d4 − 149d3 + 668d2 − 1112d + 576 + ∇.n ∆∇.n + (∇n ∇.n)2 d−1 (d − 2)(d − 1) (3d2 − 22d + 16)(d − 3) − (d − 4)(d − 6)(d − 8) ∇n3 ∇.n − (∇∇.n)2 d−1 2(13d2 − 58d + 72) − 4(d − 4)(d − 3) ∆∇n ∇.n − 3(d − 2)(d − 8) ∇n ∆∇.n + (∇.n)2 J (d − 2) 4(4d3 − 35d2 + 88d − 72) 4(d − 4)(d − 3)(d − 1) 2 − (∇n ∇.n) J + J d−2 d−2 ) − 6(3d − 4)(d − 6)∇.n ∇n J + 8(d − 3)(d − 1) ∆J + 2(d − 8)(d − 6)(d − 1) ∇n2 J .

When k = d, we have I 2 = 1 + σ d Bd¯ where the obstruction density Bd¯ = Bd¯ |Σ . Thus the following determine the obstruction density in dimensions d = 2, 3, 4 and 5. B1 = −(∇.n)2 − ∇n ∇.n − J ) 1 ( 2∆∇.n + 2 ∇n2 ∇.n + 8 ∇.n ∇n ∇.n + 3 (∇.n)3 + 8∇.n J + 8 ∇n J B2 = − 12 1 ( 9 ∇n ∆∇.n + 12 ∇.n ∆∇.n + 6 ∇.n ∇n2 ∇.n + 3 (∇∇.n)2 B3 = − 108 + 6(∇n ∇.n)2 + 18 (∇.n)2 ∇n ∇.n + 4 (∇.n)4

B4 =

) + 9 ∆J + 18 ∇n2 J + 36 ∇.n ∇n J + 18 ∇n ∇.n J + 18 (∇.n)2 J 1 ( − 312∇.n(∇n ∇.n)2 + 352∇.n(∇n3 ∇.n) − 155∇.n2 ∇.n3

46080 +44 ∇.n2 (∇n2 ∇.n) − 832∇.n3 (∇n ∇.n) + 480(∇n ∇.n)(∇n2 ∇.n)

+ 96 (∇n4 ∇.n) + 32∆∇n2 ∇.n + 288∇.n∆∇n ∇.n + 256 ∇n ∆∇n ∇.n + 96(∇a ∇.n)∇ a ∇n ∇.n − 552∇.n(∇a ∇.n)∇ a ∇.n − 608 (∇a ∇.n)∇n ∇ a ∇.n − 1436∇.n2 ∆∇.n − 1248(∇n ∇.n)∆∇.n − 2176 ∇.n∇n ∆∇.n − 864∇n2 ∆∇.n − 288∆2 ∇.n − 2304 ∇.n(∇n ∇.n)J − 832 ∇.n3 J − 640(∇n2 ∇.n)J − 384 (∆∇.n)J + 1024J2 ∇.n + 512J∇n J − 2496 ∇.n2 ∇n J − 2304(∇n ∇.n)∇n J + 1536J∇n J − 2432 ∇.n∇n2 J

) − 768 ∇n3 J − 256∆∇n J − 512 (∇a ∇.n)∇ a J − 2176∇.n∆J − 2048∇n ∆J . The above yields B1 = B1 |Σ = 0 and it is not difficult to show that B2 |Σ agrees with B2 as in Proposition 6.6. The expressions above for B3 and B4 provide an efficient way to compute the obstruction densities B3 and B4 for explicit metrics, but

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expressing the above formulæ in terms of standard hypersurface invariants is rather tedious. For (M , g) flat and d¯ = 3, this computation was performed in [15,17] and agrees with our generally curved expression for B3 given in Proposition 1.1. References [1] S. Alexakis, R. Mazzeo, Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds, Comm. Math. Phys. 297 (3) (2010) 621–651. [2] L. Andersson, P.T. Chruściel, H. Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations, Comm. Math. Phys. 149 (3) (1992) 587–612. [3] A.F. Astaneh, G. Gibbons, S.N. Solodukhin, What surface maximizes entanglement entropy? Phys. Rev. D 90 (8) (2014) 085021, arXiv:1407.4719. [4] P. Aviles, R.C. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds, Duke Math. J. 56 (2) (1988) 395–398. [5] T.N. Bailey, M.G. Eastwood, A.R. 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