Spectrum of hypersurfaces with small extrinsic radius or large λ1 in Euclidean spaces

Journal of Functional Analysis 271 (2016) 1213–1242

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Spectrum of hypersurfaces with small extrinsic radius or large λ1 in Euclidean spaces Erwann Aubry a , Jean-François Grosjean b,∗ a

LJAD, Université de Nice Sophia-Antipolis, CNRS, 28 avenue Valrose, F-06108 Nice, France b Institut Élie Cartan de Lorraine (Mathématiques), Université de Lorraine, site de Nancy, B.P. 70239, F-54506 Vandœuvre-les-Nancy cedex, France

a r t i c l e

i n f o

Article history: Received 23 January 2015 Accepted 20 June 2016 Available online 27 June 2016 Communicated by L. Saloff-Coste MSC: 53A07 53C21 Keywords: Extremal hypersurfaces Spectrum

a b s t r a c t The Reilly and Hasanis–Koutroufiotis inequalities give sharp bounds on λ1 and on the extrinsic radius of Euclidean hypersurfaces in terms of the L2 norm of their mean curvature. The equality case of these inequalities characterizes the Euclidean spheres. In this paper, we study the spectral properties of the almost extremal hypersurfaces. We prove that the spectrum of the limit sphere asymptotically appears in the spectrum of almost extremal hypersurfaces for these inequalities. We also construct some examples of extremizing sequences that prove that the limit spectrum can be essentially any closed subset of R+ that contains the spectrum of the limit sphere. We also provide natural sharp condition to recover exactly the spectrum of the unit sphere. © 2016 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (E. Aubry), [email protected] (J.-F. Grosjean). http://dx.doi.org/10.1016/j.jfa.2016.06.011 0022-1236/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction Throughout the paper, X: M n → Rn+1 is a closed, connected, immersed Euclidean hypersurface (with n  2). We let vM be its volume, BM its second fundamental form, HM = n1 tr BM its mean curvature, rM its extrinsic radius (i.e. the least radius of the M M Euclidean balls containing M ), 0 = λM 0 < λ1  λ2  · · · the non-decreasing sequence of its eigenvalues labeled with multiplicities, Sp(M ) = (λM i )i∈N and its center of mass. ⎛ ⎞ α1  For any function f : M → R, we set f α = ⎝ v1M |f |α dv ⎠ . M

The Hasanis–Koutroufiotis inequality ([8], see also section 3 of this paper) and the Reilly inequality ([9], see also section 3 of this paper) assert respectively that {1  rM HM 2 } and



 2 λM 1  nHM 2 ,

(1.1)

with equality in one of these inequalities if and only if M is a Euclidean sphere (which is then uniquely determined). Our aim is to study the spectral properties of the hypersurfaces that are almost extremal for at least one of the inequalities (1.1). In the sequel, for any immersed hyper1 1 and center X := Xdv. surface M → Rn+1 , we let SM be the sphere of radius H 2 vM M

It follows from the above-mentioned results of Hasanis–Koutroufiotis and Reilly that equality holds in one of the inequalities in (1.1) if and only if M = SM . For any k  0, we let μSk M := k(n+k−1)H22 be the k-th eigenvalue of SM (labeled without multiplicities) and mk be its multiplicity. Our first result is the following Theorem 1.1. We fix n  2 and τ > 0. Then, there exists ε0 (n, k, τ ) > 0 depending only on n, τ and k such that for any ε < ε0 (n, k, τ ) and any immersed hypersurface M → Rn+1 satisfying either {1  rM H2  1 + ε} or



2 M λM 1  nH2  (1 + ε)λ1



(1.2)

then the interval [(1 − τ )μSk M , (1 + τ )μSk M ] contains at least mk eigenvalues of M counted with multiplicities. We will see in the proof that ε0 (n, k, τ ) tends to 0 when k → ∞ or τ → 0. Note that almost extremal hypersurfaces for the Reilly inequality must have at least n + 1 eigenvalues close to λS1 M = nH22 . However, they can have the topology of any immersed hypersurface of Rn+1 (see below) and can be as close as wanted of any closed, connected subset of Rn+1 that contains Sn (see [4]). So almost extremal hypersurfaces for the Reilly inequality are very different from almost extremal manifolds for the Lichnerowicz Inequality in positive Ricci curvature (see for instance [2]).

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Now for any sequence (Mk )k∈N , let us define LimSet Sp(Mk ) := k→∞



Sp(M ).

k∈N k

This is the union of the limit-sets of all the sequences (μk )k∈N with μk ∈ Sp(Mk ) for any k ∈ N. Obviously if (Sp(Mk ))k converges to a set F for the Attouch–Wetts–Hausdorff distance (see section 2 below for the definition), then LimSet Sp(Mk ) = F . As a consek→∞

quence of Theorem 1.1, we have Corollary 1.2. Let (Mk )k∈N be a sequence of immersed hypersurfaces of Rn+1 normalized by HMk 2 = 1 and such that  either

lim rMk = 1

k→∞

 or

k lim λM = n . 1

k→∞

(1.3)

Then we have LimSet Sp(Mk ) ⊃ Sp(Sn ). k→∞

 In other words, F ⊃ Sp(Sn ) for any limit-point F of the sequence Sp(Mk ) k∈N for the Attouch–Wetts–Hausdorff distance. Conversely, our result is optimal in the sense that any closed set containing the spectrum of a Euclidean sphere can be achieved as the spectrum of an “almost extremal” manifold. This is the object of our second result: Theorem 1.3. Let M → Rn+1 be any immersed hypersurface. Let F be any closed subset such that Sp(Sn ) ⊂ F ⊂ [0, +∞[. Then there exists a sequence (ik )k of immersions ik : M → Rn+1 such that, denoting Mk := ik (M ), it satisfies lim rMk HMk 2 = 1 and lim Sp(Mk ) = F

k→+∞

k→∞

for the Attouch–Wetts–Hausdorff distance. If moreover we have F ⊂ {0} ∪ [n, +∞[ then k λM 1 we can obtain lim = 1. The sequence of immersions ik : M → Rn+1 is k→+∞ nHMk 2 such that HMk 2 = 1 and lim vMk = vSn . In addition, we have the following curvature k→∞ properties 

 lim

k→∞ Mk

for any 1  α < n.

|BMk | dv =

|BSn |α dv

α

Sn

(1.4)

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Theorem 1.3 is a special case (M1 = Sn and M2 = M ) of the more general Theorem 2.1 of Section 3. Remark 1.4. In the case α = n, we are only able to get a weak version of Theorem 1.3 with F = Sp(M1 ) ∪ G, where G is a finite set whose elements are known up to an error  n term and such that lim |B| dv is bounded above by a constant that depends on M , Mk

on the cardinal of G, on the distance between G and Sp(M1 ), and on the error term. We now investigate a natural condition on “almost extremal” manifolds to rule out the formation of a non-spherical spectrum. As proved by the authors in [4], any “almost extremal” hypersurface is arbitrary Hausdorff-close to its spherical model provided an Lα -control (α > n) on the second fundamental form. This result combined with the C 1,β pre-compactness theorem of [7] (or a Moser iteration as in [3]) implies the following stability in Lipschitz distance dL : Theorem 1.5. We fix α ∈ (n, +∞], A > 0 and τ > 0. Then there exists ε0 (n, α, A, τ ) > 0 depending only on n, α, A and τ such that for any ε < ε0 and any immersed hypersurface M → Rn+1 satisfying either {1  rM H2  1 + ε} or



2 M λM 1  nH2  (1 + ε)λ1



and vM Bnα  A, then M is diffeomorphic to SM and satisfies dL (M, SM ) < τ . Moreover there exists ε1 (n, k, α, A, τ ) > 0 depending only on n, k, α, A and τ such SM that if ε < ε1 then |λM k − λk |  τ . Therefore, Theorem 1.3 is optimal in the sense that it is enough to improve slightly (1.4) to get convergence to the spectrum of the sphere. In the following theorem proved in [3], we construct almost extremal hypersurfaces for the Hasanis–Koutroufiotis inequality, not diffeomorphic to SM , Gromov–Hausdorff close to SM , with H∞ bounded, where the limit spectrum is that of Sn . But the number of eigenvalues of M close to each eigenvalue μk of Sn is a multiple of the multiplicity mk . Theorem 1.6. For any integers l, p there exists sequence of embedded hypersurfaces (Mj )j of Rn+1 diffeomorphic to p spheres Sn glued by connected sum along l points, such that Hj ∞  C(n), Bj n  C(n), rMj → 1, Hi 2 → 1, and for any σ ∈ N we have λσ (Mj ) → λE( σp ) (Sn ). In particular, the Mj have at least pmk eigenvalues close to μk .

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The structure of the paper is as follows: in section 2 we state the Theorem 2.1 which is a general construction which gives the Theorem 1.3. After a preliminary section 3, where we give short proofs of the Reilly and Hasanis–Koutroufiotis inequalities, we prove in section 4 some concentration properties for the volume, the mean curvature and the position vector X for almost extremal hypersurfaces. Section 5 is devoted to estimate on the restriction to hypersurfaces of the homogeneous, harmonic polynomials of Rn+1 . These estimates are used in Section 6 to prove Theorem 1.1. We end the paper in section 7 by the proof of the constructions of Theorem 2.1. The results and estimates of this paper are used in [4] to study the metric shape of the almost extremal hypersurfaces. Notations: Note that throughout the paper we adopt the notation that C(n, k, p, · · · ) is function greater than 1 which depends on p, q, n, · · · . It eases the exposition to disregard the explicit nature of these functions. The convenience of this notation is that even though C might change from line to line in a calculation it still maintains these basic features. Note that all these constants are computable. For convenience, we will often write B = BM , H = HM , and more generally we will drop the index M in the geometric quantities. Acknowledgments: Part of this work was done while E.A was invited at the MSI, ANU Canberra, funded by the PICS-CNRS Progress in Geometric Analysis and Applications. E.A. thanks P. Delanoe, J. Clutterbuck and J.X. Wang for giving him this opportunity. This paper was partially funded by the ANR-10-BLAN-0105 (ANR ACG). The authors are grateful to the anonymous referee for his/her very constructive remarks that helped improve the presentation of the paper. 2. Miscellaneous on Theorems 1.1 and 1.3 We will prove the general construction Theorem 2.1 below. As mentioned in the introduction, Theorem 1.3 is an immediate consequence of Theorem 2.1. Theorem 2.1. Let M1 , M2 → Rn+1 be two immersed compact submanifolds of dimension m  3, M1 #M2 be their connected sum and F be any closed subset of (0, +∞) containing Sp(M1 ). Then there exists a sequence of immersions ik : M1 #M2 → Rn+1 with induced metric gk and volume vk such that (1) ik (M1 #M2 ) converges to M1 in Hausdorff topology, (2) the curvatures of gk satisfy ⎧ 1 ⎪ lim ⎪ ⎪ ⎪ k→∞ v k ⎨ 1 ⎪ ⎪ lim ⎪ ⎪ ⎩ k→∞ vk

 |B|α dv = ik (M1#M2 )

|H| dv = α

ik (M1 #M2 )

1 vM1 1 vM1

 |B|α dv

for any 1  α < m

|H|α dv

for any 1  α < m

M 1

M1

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 (3) lim Sp ik (M1 #M2 ) = F , for the Attouch–Wetts–Hausdorff distance, k→∞

(4) lim vk = vM1 . k→∞

Remark 2.2. In the case α = m, we are only able to get a weak version of Theorem 2.1 with F = Sp(M1 ) ∪ G, where G is a finite set whose elements are known up to an error  |B|m dv is bounded by a constant

term and where the point (2) is replaced by ik (M1 #M2 )

that depend on M1 , M2 , on the cardinal of G, on the distance between G and Sp(M1 ) and on the error term. Now we recall the definition of the Attouch–Wetts–Hausdorff distance for the sets of R. If dA : R → R denotes the distance function to the subset A, we have dH (A, B) = dA − dB ∞ and so the Hausdorff topology on compact subsets of R coincides with the topology of the uniform convergence on R of the associated distance functions. Seemingly, on the set of closed subset of R we consider the Attouch–Wetts topology, that is the topology of the uniform convergence on compact subsets of the distance functions. It is a complete, metrizable topology induced by the distance dAW (A, B) =



 2−N inf 1, sup |dA (x) − dB (x)|

N ∈N∗

x∈[0,N ]

We have lim dAW (Ak , B) = 0 if and only if lim dN (Ak , B) = 0 for any N ∈ N large k

k

enough, where dN (A, B) = inf{ε > 0 | A ∩ [0, N ] ⊂ Bε et B ∩ [0, N ] ⊂ Aε } and Aε := {x ∈ R | d(x, A)  ε} (see the proof of Proposition 3.1.6 in [5]). In the proof of Theorem 2.1, we will need of the following construction. If F is a closed subset of R, there exists an increasing sequence of finite sets FN := {x1 , · · · , xkN } such  kN 

1 1 x i − , xi + = FN,1/N . In this case we can easily that FN ⊂ [0, N ] ∩ F ⊂ N N i=1 prove that lim dAW (FN , F ) = 0 and F = LimSetFN . N →∞

N →∞

3. Some geometric optimal inequalities Any function F on Rn+1 gives rise to a function F ◦ X on M which, for more convenience, will be also denoted F subsequently. If Δ denotes the Laplace operator of (M, g), then we have ΔF = nHdF (ν) + Δ0 F + ∇0 dF (ν, ν),

(3.1)

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where ν denotes a local normal vector field of M in Rn+1 , ∇0 is the Euclidean connection and Δ0 is the Laplace operator of Rn+1 . Applied to F (x) = x − X, x − X , where · , · is the canonical product on Rn+1 , Formula (3.1) gives the Hsiung formulae,    1 Δ|X − X|2 = nH ν, X − X − n, H ν, X − X dv = vM . (3.2) 2 M

3.1. A rough geometrical bound The integrated Hsiung formula (3.2) and the Cauchy–Schwarz inequality give the following    H ν, X − X dv 1=  H2 X − X 2 (3.3) vM M

  This inequality H2 X − X2  1 is optimal since M satisfies H2 X − X 2 = 1 1 and center X. Indeed, in this case X − X if and only if M is a sphere of radius H 2

and ν are collinear on M \ {H = 0}, hence |X − X|2 (and so H) is locally constant on M \ {H = 0}. By connectedness and compactness of M , this implies that H is constant and non-zero on M . {H = 0} = ∅ and that X is an isometric-cover of M on the sphere ¯ 2 = 1 , hence an isometry. S of center X and radius X − X H2

3.2. Hasanis–Koutroufiotis inequality on extrinsic radius We set R the extrinsic Radius of M , i.e. the least radius of the balls of Rn+1 which contain M . Then Inequality (3.3) gives H2 rM = H2 inf u∈Rn+1 X − u∞  1 H2 infn+1 X − u2 = H2 X − X2  1 and rM = H if and only if we have 2 u∈R

equality in (3.3). 3.3. Reilly inequality on λM 1  Since we have

1 vM

¯ by the (Xi − X¯i ) dv = 0 for any component function of X − X,

M  1 M M ¯ 2 min–max principle and Inequality (3.3), we have λM Xi − 1 H22  λ1 X−X2 = λ1 i  X¯i 22  ∇Xi 22 = n where λM 1 is the first non-zero eigenvalue of M and where the i  last equality comes from the fact that |∇Xi |2 is the trace with respect to the canonical i

scalar product of the quadratic form Q(u) = |p(u)|2 , where p is the orthogonal projector from Rn+1 to Tx M . This gives the Reilly inequality in (1.1). Here also, equality in the Reilly inequality gives equality in (3.3) and so it characterizes  1 n the sphere of radius H2 = X2 = λM . 1

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4. Concentration estimates In the section, we prove that almost extremal hypersurfaces are close to a sphere and have almost constant mean curvature in L2 -norm. We say that M satisfies the pinching (Pp,ε ) when Hp X − X2  1 + ε. From the proofs of Inequalities (1.1) above, it appears that pinchings rM H2  1 + ε or nH22 /λ1  1 + ε imply the pinching (P2,ε ). In all the results of this section, we have 0 < ε < 1. ¯ = 0. Let X T (x) denote From now on, we assume, without loss of generality, that X the orthogonal projection of X(x) on the tangent space Tx M . In the following lemma, we see that the position vector X almost satisfies, in L2 -norm, characteristics properties of the Euclidean spheres (X T = 0). √ H Lemma 4.1. If (P2,ε ) holds, then we have X T 2  3εX2 and X − H 2 ν2  2 √ 3εX2 .  1 Proof. Since we have 1 = vM H X, ν dv  H2  X, ν 2 , Inequality (P2,ε ) gives us M

X2  (1 + ε) X, ν 2 and 1  H2 X2  1 + ε. Hence X − X, ν ν2  −2 Hν 2 2 and X − H  3ε X22 . 2 2 2 = X2 − H2

√

3ε X2

2

In the lemma below, we see that in L2 -norm, M is close to a sphere and has L2 -almost constant mean curvature. In particular, the volume of M is concentrated in a tubular 1+η 1−η neighborhood Aη of the sphere SM where Aη := B0 ( H ) \ B0 ( H ) for some η. 2 2 2 M Lemma 4.2. If (Pp,ε ) (for p > 1 + ε, or rM H2  1 + ε holds  2), or 1nH  2 /λ1C  √ √ 1 (with ε  100 ), then we have |X| − H2 2  H2 8 ε, |H| − H2 2  C 8 εH2 and 2p √ 8 8 ε)  C Vol (M \ A √ εvM , where C = 6 × 2 p−2 in the case (Pp,ε ) and C = 100 in the other cases.

Proof. When (Pp,ε ) holds, we have 2 1− p

p Hp X2  (1 + ε)  (1 + ε)Hp X p−1  (1 + ε)Hp X1

  1 2 1 hence we get |X| − H = X22 − 2 X H2 + 2 2 second inequality of Lemma 4.1, it gives

1 H22

2

X2p ,

2p

1  2 p−2 H 2 ε. Combined with the 2

      √ |H| − H2   H22 |X| − |H|  + H22 |X| − 1   C 4 εH2 2 H22 2 H2 2 Now, by the Chebyshev inequality and Lemma 4.1, we get  

 4 ε Vol M \ A √ = Vol x ∈ M/ |X(x)| −

√ 4 1  ε   H2 H2

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

H2  √ 2 ε



 |X| −

1221

√ 1 2 dv  C(p) εvM H2

M

When rM H2  1 + ε holds. We set X0 the center of the circumsphere to M of 2 2 radius rM . We have X − X0 22 = X22 + |X0 |2 = rM  (1+ε) and then we have H2 √

√

√ 4 4 ε) Vol (M \ A √ 1 ε  vM H22 vM 1  vM



 M \A



√ 4ε

1  2 = 1 H2 2 vM √



 M ∩A

2

 1 2 − |X| dv H22



√ 4ε

where in the last inequality we have used |X|   |X| −

1 − |X|2 ∈ [ Hε2 , H ] on 2

1 H22

1 |X| − H22



2

M ∩A

 |X| − √ 4ε

√ 4

2

ε 3ε |X0 |  H and |X|  |X0 | + rM  1+3 H2 . So we have 2 4 ε . Chebyshev inequality and (3.3) give us M \A√

√ 1+3 ε H2

dv 

√ 9 ε H22

and, so we get

1 2 1 dv + H2 vM

 M \A

√ 4

 |X| − √ 4ε

1 2 dv H2

4 ε) Vol (M \ A √ 10 ε ε 1 +  2 2 H2 vM H2 H22

  √  1 |H|  C 8ε Combined with the second inequality of Lemma 4.1, we get  H − 2   H . H 2 2 2 2 

2  2 2 M When nH2 /λ1  1 + ε holds, we have |X| − X2 dv = 0 and so by the M

 2 4X T 2 12(1+ε)2 εX22 Poincare inequality we get |X|2 − X22 2  λM 2   2 1     nH 2  2  2 1  1  1 2 2   |X| − X2  + X2 − |X| − gives  |X| − 2 H2

H2

H2

2

2

2

200ε , which nH42  12√ε 1   H2 H22 2

4 ε by the same Chebyshev procedure as and then we get the estimate on the volume of A √ for Pp,ε and the estimate on the mean curvature by the same procedure as for rM H2  1 + ε. 2

For our last estimates, we some notations. a smooth  need  Let ψ:[0, ∞) →16√[0,2 1] be 16√ √ √ 2 (1−2 16 ε)2 (1+2 16 ε)2 (1− ε) (1+ function with ψ=0 outside , H2 and ψ=1 on [ H2 , H2ε) ]. Let H2 2

2

2

2

us consider the function ϕ on M defined by ϕ(x) = ψ(|Xx |2 ) and the vector field Z on M defined by Z = ν − HX. For any sphere RSn , Z is vanishing. The previous estimates then imply the following lemma and we see that in L2 -norm, Z is small.  2 2 H − Lemma 4.3. (P ) (for p > 2) or nH /λ  1 + ε or r H  1 + ε implies p,ε 1 M 2 2  √ √ 3 2 2 2 8 8 H2 1  C εH2 , Z2  Cε 32 and |ϕ2 − 1|  C ε, where C is a constant which depends on p in the case (Pp,ε ).

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    √ Proof. We have H2 − H22 1  2|H| − H2 2 H2  C 8 εH22 and Z22 =

1 vM

 |Z|2 dv = M

=

H22



vM

1 vM

 1 − 2H ν, X + H2 |X|2 dv M

 X −

M

H 2 1 ν dv + H22 H22 vM

 (H22 − H2 )(1 − |X|2 H22 )dv M

   √ H2 − H22 1 H  2 2 16  H2 X − ν + 8 ε , H22 2 H22 which gives the result by Lemma 4.1. Finally, we have 1 − ϕ22 and ϕ22  1. 2

Vol (M \A vM

√ 8 ε)



Vol (A √ 8 ε ∩M ) vM



5. Homogeneous, harmonic polynomials of degree k The eigenfunctions of Sn are restrictions to Sn of homogeneous, harmonic polynomials of the ambient space Rn+1 . To prove Theorem 1.1, we will use restrictions to M of homogeneous, harmonic polynomials as quasi-modes. In that purpose, we prove in this section, some estimates on harmonic homogeneous polynomials and their restrictions to Euclidean hypersurfaces. 5.1. General estimates Let Hk (Rn+1 ) be the space of homogeneous, harmonic polynomials of degree k on n Rn+1 . Note that Hk (Rn+1 ) induces on Sn the spaces of eigenfunctions of ΔS associated  n + k − 1 n + 2k − 1 . to the eigenvalues μk := k(n + k − 1) with multiplicity mk := k n+k−1  1 On the space Hk (Rn+1 ), we set (P, Q)Sn := P Qdvcan , where dvcan denotes the vSn Sn

element volume of the sphere with its standard metric. Remind that for any P ∈ Hk (Rn+1 ) and any X, Y ∈ Rn+1 , we have the Euler identities dX P (X) = kP (X) and ∇0 dX P (X, Y ) = (k − 1)dX P (Y ).

(5.1)

Lemma 5.1. For any x ∈ Rn+1 and P ∈ Hk (Rn+1 ), we have |P (x)|2  P 2Sn mk |x|2k . Proof. Let (Pi )1imk be an orthonormal basis of Hk (Rn+1 ). For any x ∈ Sn , Qx (P ) = mk  P 2 (x) is a quadratic form on Hk (Rn+1 ) whose trace is given by Pi2 (x). Since for any i=1

x ∈ Sn and any O ∈ On+1 such that x = Ox we have Qx (P ) = Qx (P ◦ O) and since

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

mk  Pi2 (x) = tr (Qx ) = Pi2 (x ) = i=1 i=1   mk 1 = mk = Pi2 (x) dvcan and so vSn i=1

P → P ◦ O is an isometry of Hk (Rn+1 ), we have  mk  1 tr (Qx ). We infer that Pi2 (x)dvcan n v S i=1 mk 

Sn

1223

mk 

Sn

Pi2 (x) = mk . By homogeneity of the Pi we get

i=1 mk 

Pi2 (x) = mk |x|2k ,

(5.2)

i=1

and by the Cauchy–Schwarz inequality applied to P (x) =



(P, Pi )Sn Pi (x), we get the

i

result. 2

As an immediate consequence, we have the following lemma. Lemma 5.2. For any x, u ∈ Rn+1 and P ∈ Hk (Rn+1 ), we have |dx P (u)|2  P 2Sn mk

!μ

"

μk  |x|2(k−1) |u|2 + k2 − u, x 2 |x|2(k−2) . n n k

Proof. Let x ∈ Sn and u ∈ Sn so that u, x = 0. Once again the quadratic forms

2 Qx,u (P ) = dx P (u) are conjugate (since On+1 acts transitively on orthonormal coumk  2

ples) and so dx Pi (u) does not depend on u ∈ x⊥ nor on x ∈ Sn . By choosing an i=1

orthonormal basis (uj )1jn of x⊥ , we obtain that mk 

i=1

mk  n 2

2 1 1 dx Pi (u) = dx Pi (uj ) = n i=1 j=1 nvSn

=

1 nvSn

  mk

Pi ΔS Pi dvcan = n

Sn i=1

  mk

|∇S Pi |2 dvcan n

Sn i=1

mk μk n

Now suppose that u ∈ Rn+1 . Then u = v + u, x x, where v = u − u, x x, and we have mk 

mk 2 

2 dx Pi (u) = dx Pi (v) + k u, x Pi (x)

i=1

i=1

=

mk 

i=1

mk  2 dx Pi (v)Pi (x) + mk u, x 2 k2 dx Pi (v) + 2k u, x i=1

!μ " ! mk μk 2 μk " k = |v| + mk u, x 2 k2 = mk |u|2 + k2 − u, x 2 , n n n

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where we have taken the derivative the equality (5.2) to compute homogeneity of Pi we get

mk 

dx Pi (u)

2

= mk

μk n

mk  dx Pi (v)Pi (x). By i=1

|x|2(k−1) |u|2 +(k 2 − μnk ) u, x 2 |x|2(k−2)



i=1

and conclude once again by the Cauchy–Schwarz inequality. 2 Lemma 5.3. For any x ∈ Rn+1 and P ∈ Hk (Rn+1 ), we have |∇0 dP (x)|2  P 2Sn mk αn,k |x|2(k−2) , where αn,k = (k − 1)(k2 + μk )(n + 2k − 3)  C(n)k4 . Proof. The Bochner equality gives mk 

|∇ dPi (x)| = 0

2

i=1

mk   i=1

1  2 dΔ Pi , dPi − Δ0 dPi  2



0

 1 = − mk k2 + μk Δ0 |X|2k−2 = mk αn,k |X|2k−4 . 2

2

(5.3)

5.2. Estimates on hypersurfaces The main result of this section is the Lemma 5.6 which is fundamental in the proof of Theorem 1.1. It controls the defect of the localized restriction map P ∈ Hk (Rn+1 ) → Hk2 ϕP ◦ X ∈ L2 (M ) to be an isometry. Note that it applies to any Euclidean hypersurface. In the case of almost extremal hypersurface, it will prove that the localized restriction map is a quasi-isometry (see Lemma 6.1). Let Hk (M ) = {P ◦ X , P ∈ Hk (Rn+1 )} be the space of functions induced on M by k H (Rn+1 ). We will identify P and P ◦ X subsequently. There is no ambiguity since we have Lemma 5.4. Let M n be a compact manifold immersed by X in Rn+1 and let (P1 , . . . , Pm ) be a linearly independent set of homogeneous polynomials of degree k on Rn+1 . Then the set (P1 ◦ X, . . . , Pm ◦ X) is also linearly independent. Proof. Any homogeneous polynomial P which is zero on M is zero on the cone R+ ·M . Since M is compact there exists a point x ∈ M so that Xx ∈ / Tx M and so R+ ·M has non-empty interior. Hence P ◦ X = 0 implies P = 0. 2 We first need to precise the localization functions ϕ for which Lemma 5.6 applies. Let 0 < η < 1 be fixed. We ψ : [0, ∞) −→ [0, 1] a smooth function which  still denote  2 (1+η)2 (1−η/2)2 (1+η/2)2 is 0 outside (1−η) , is 1 on , H2 , and satisfies the upper bounds H2 H2 H2 2

|ψ  | 

4H22 η

and |ψ  | 

2

8H42 η2 .

2

2

We set ϕ(x) = ψ(|Xx |2 ) on M .

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Lemma 5.5. With the above restrictions on ψ we have Δϕ2 1 

192H42 16nH22 T 2 Z1 . X  + 2 η2 η

Proof. An easy computation yields that Δ(ϕ2 ) = −(ψ 2 ) (|X|2 )|d|X|2 |2 + (ψ 2 ) (|X|2 )Δ|X|2 = −4(ψ 2 ) (|X|2 )|X T |2 − 2n(ψ 2 ) (|X|2 ) ν, Z But the bound on the derivatives of ψ gives us |(ψ 2 ) |  Hence we get Δϕ2 1 

192H42 X T 22 η2

+

16nH22 ϕZ1 . η

8H22 η ψ

and |(ψ 2 ) | 

48H42 . η2

2

Lemma 5.6. Let ϕ : M → [0, 1] be as above. There exists a constant C = C(n) such that for any isometrically immersed hypersurface M of Rn+1 and any P ∈ Hk (M ), we have k   2  2 2 2 H2k  DC(n) P  ϕP  − ϕ P  mi (1 + η)2i n n 2 2 2 S

S

i=1

where D = Z2 + Z22 +

200H22 X ⊥ 22 η2

+

16n η Z1

+

H2 −H22 1 H22

and Z = ν − XH.

Proof. For any P ∈ Hk (M ), the Euler identities (5.1) give us ϕ∇0 P 22 = ϕdP (ν)22 + ϕ∇M P 22 = ϕdP (Z)22 + ϕdP (HX)22  

2 1 1 + 2ϕ HdP (Z)dP (X) + ∇M ϕ2 P, ∇M P − ∇M ϕ2 , ∇M P 2 dv vM 2 M

= ϕdP (Z)22 + k2 ϕHP 22 +

1 vM



 1 2kHdP (ϕZ)ϕP + ϕ2 P ΔP − P 2 Δ(ϕ2 ) dv 2

M

Now, Formula (3.1) applied to P ∈ Hk (Rn+1 ) gives ΔP = μk H2 P + (n + 2k − 2)HdP (Z) + ∇0 dP (Z, Z)

(5.4)

hence, we get ϕ∇0 P 22 = dP (ϕZ)22 + (μk + k2 )HϕP 22 

2  1 1 ϕ P ∇0 dP (Z, Z) + (n + 4k − 2)ϕHdP (ϕZ)P − P 2 Δ(ϕ2 ) dv + vM 2 M

1 = vM

 ! "

 (μk + k2 ) H2 − H22 ϕ2 P 2 + (n + 4k − 2)HdP (ϕZ)ϕP dv M

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1226

+

1 vM



 1 P ∇0 dP (ϕZ, ϕZ) − P 2 Δ(ϕ2 ) dv 2

M

+ (μk + k2 )H22 ϕP 22 + dP (ϕZ)22 Now we have  n 2  0 2  2 2 ∇ P  n =  ∇S P  + k2 P Sn = (μk + k2 ) P Sn S Sn

(5.5)

Hence 2  H22k−2 ϕ∇0 P 22 − ϕ22 ∇0 P Sn

2  2k−2 2 2 = (μk + k2 ) H2k dP (ϕZ)22 2 ϕP 2 − ϕ2 P Sn + H2  ! "

 H22k−2 + ϕ2 P (μk + k2 ) H2 − H22 P + H(n + 4k − 2)dP (Z) + ∇0 dP (Z, Z) dv vM M

−

H22k−2



vM

1 2 P Δ(ϕ2 )dv 2

M

Which gives    2  2 2 H2k 2 ϕP 2 − ϕ2 P Sn      1  2k−2 0 2 2  0 2  ∇  ϕ∇ P  − ϕ P H 2 2 2 Sn  μk + k2  " H22k−2 ! 2 0 2 + + |P ||∇ dP ||ϕZ| (n + 4k − 2)|H|ϕ|P ||dP (ϕZ)| + |dP (ϕZ)| μk + k2 M

+

H22k−2



vM

   P2 |Δ(ϕ2 )| dv ϕ2 H2 − H22 P 2 + 2

(5.6)

M

By Lemma 5.1, we have H22k−2 vM



2 2k−2  2  H − H22 (ϕP )2 dv  mk P Sn H2 vM

M



  2 2 ϕ (H − H22 )|X|2k dv

M 2

 P Sn mk (1 + η)2k

ϕ2 (H2 − H22 )1 H22

In the same way, we have H22k−2 vM

 M

P2 Δϕ2 1 2 |Δ(ϕ2 )|dv  P Sn mk (1 + η)2k 2 H22

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1227

and using Lemma 5.2, we get H22k−2 vM

 2

ϕ2 |P dP (Z)H|dv  P Sn mk k(1 + η)2k ϕ2 Z2 M

and H22k−2 vM

 |dP (ϕZ)|2  P 2Sn mk k2



H22k−2 vM

M

|ϕZ|2 |X|2(k−1) dv M



2 P Sn

mk k (1 + η) ϕZ22 2

2k

Finally, using Lemma 5.3, we get H22k−2 vM

 |P ||∇ dP ||ϕZ|  0

2

H22k−2 √ P 2Sn mk αn,k vM

M



2 P Sn

 |X|2(k−1) |ϕZ|2 dv M

√

mk αn,k (1 + η)2k ϕZ22

which, combined with (5.6) and equation (5.5), gives  2  2 2 H2k 2 ϕP 2 − ϕ2 P  n P 2Sn

S



 2    H22k−2 ϕ∇0 P 22 − ϕ22 ∇0 P Sn  2

∇0 P Sn

! Δ(ϕ2 )1 ϕ2 (H2 − H22 )1 " + C(n)mk (1 + η)2k ϕ2 Z2 + ϕZ22 + + H22 H22     2   H22k−2 ϕ∇0 P 22 − ϕ22 ∇0 P Sn  + C(n)mk (1 + η)2k D  2 ∇0 P Sn where we have used the previous lemma. Since in case k = 1, |∇0 P | is constant we get   H22 ϕP 22 − ϕ22 P 2n   C(n)m1 (1 + η)2 D P 2n S

S



 2 2 2 |H2k k n+1 2 ϕP 2 −ϕ2 P Sn | Now, let Bk = sup | P ∈ H (R ) \ {0} . Then using that 2 P Sn # k n+1 0 2 2 k−1 for any P ∈ H (R ), we have |∇ P | = i |∂i P | with ∂i P ∈ H (Rn+1 ), we get       H2k−2 ϕ∇0 P 22 − ϕ22 ∇0 P 2n   H2k−2 ϕ∂i P 22 − ϕ22 ∂i P 2n  2 2 S S i

 Bk−1



2  2 ∂i P Sn = Bk−1 ∇0 P Sn

i

and by (5.5), it gives Bk  Bk−1 + C(n)mk (1 + η)2k D  C(n)D

#k i=1

mi (1 + η)2i .

2

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6. Proof of Theorem 1.1 To prove Theorem 1.1 we will show (Lemma 6.3) that for extremal hypersurfaces M , the functions ϕP are almost eigenfunctions of M in L2 sense. The estimates of Lemmas 6.3 and 6.1 need to be compared to the fact proved in proved in [4] that the limit set for the Hausdorff distance of an extremizing sequence of hypersurfaces can contain any closed, connected subset of Rn+1 that contains Sn . Under the assumption of Theorem 1.1, we can use Lemma 4.1, Lemma 4.3 and the √ pinching P2,ε to improve the estimate in Lemma 5.6 in the case η = 2 16 ε. Lemma 6.1. For any isometrically immersed hypersurface M → Rn+1 with rM H2  1 + ε (or λ1 (1 + ε)2  nH22 or (Pp,ε ) for p > 2) and for any P ∈ Hk (M ), we have  √ 2  2 2 32 H2k ε P Sn , 2 ϕP 2 − P Sn  C where C = C(n, k) in the first two cases and C = C(p, k, n) in the latter case. Note that C tends to infinity when k tends to infinity. As a consequence, the map P → ϕP is injective on Hk (M ) for ε small enough and is a quasi-isometry. Lemma 6.2. Under the assumption of Lemma 6.1, if ε 

1 (2C)32

then dim(ϕHk (M )) = mk .

Lemma 6.1 allows us to prove the following estimate on ΔP , which says that for extremal hypersurfaces, ϕP is in L2 -norm an almost eigenfunction on M . 1 Lemma 6.3. Under the assumptions of Lemma 6.1, if ε  (2C) 32 , then for any P ∈   √ S S Hk (M ), we have Δ(ϕP ) − μk M ϕP 2  C 16 εμk M ϕP 2 where C = C(n, k) (C = C(n, k, p) under the pinching (Pp,ε )).

Proof. Let P ∈ Hk (M ). Using (3.1) we have Δ(ϕP ) = P Δϕ − 2 dP, dϕ + ϕΔP = P Δϕ − 2 dP, dϕ + ϕnHdP (ν) + ϕ∇0 dP (ν, ν) = P Δϕ − 2 dP, dϕ + ϕμk |H|H2 P + ϕ(n + k − 1) + ϕ(n + k − 1)

H H2 dP (Z) |H|

H (|H| − H2 )dP (ν) + ϕ∇0 dP (ν, Z) |H|

hence, we get Δ(ϕP ) − μk H22 ϕP 2  (Δϕ)P 2 + 2 dϕ, dP 2 + μk (|H| − H2 )ϕP 2 H2   + (n+k−1)H2 ϕ|dP ||Z|2 + (n+k−1)ϕ(|H| − H2 )dP (ν)2 + ϕ|∇0 dP ||Z|2 (6.1)

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1229

Let us estimate (Δϕ)P 2 . (Δϕ)P 22

1  vM



(4|ψ  (|X|2 )||X T |2 + 2n|ψ  (|X|2 )||Z|)2 P 2 dv

M

mk !  vM



2 " 2 |X|2k 4|ψ  (|X|2 )||X T |2 + 2n|ψ  (|X|2 )||Z| dv P Sn

M

√  mk (1 + 2 16 ε)2k !  vM H2k 2 

8H42 T 2 2H22 2 " √ √ |Z| dv P 2Sn |X | + 2n 16 8 ε ε

A2

√ 16 ε

A2

√ 16 ε



√  mk (1 + 2 16 ε)2k ! vM H2k 2

 " 4 128H82 T 4 2 2 H2 2 √ √ dv P Sn | + 32n |X |Z| 4 8 ε ε

Sincewe have |X T |  |X| and since Lemma 4.3 is valid with ϕZ22 replaced by 1 |Z|2 dv, we get vM A2

√ 16 ε



2

(Δϕ)P 22

C(n, k)μk P Sn  vM H2k 2 

A2

H62 T 2 H42 2  √ |X | + √ |Z| dv 4 8 ε ε

√ 16 ε

√ C(n, k)μk 2 H42 16 ε P Sn 2k H2

From the Lemma 6.1, ε 

1 (2C)32

implies that 2

2 P Sn  2H2k 2 ϕP 2

(6.2)

which gives (Δϕ)P 22  C(n, k)μk H42

√

16

εϕP 22

(6.3)

Now  dϕ, dP 22



 4ψ (|X| )|X 2

T

||dP |22

√

16

√ 16 ε

|X T |2 mk nk 2 |X|2(k−1) dv A2

 C(n, k)μk

|X T |2 |dP |2 dv A2



16H42 2 √ P Sn  16 εvM



16H42 √  16 εvM

√ 16 ε

2

εH24−2k P Sn  C(n, k)H42

√

16

εϕP 22

(6.4)

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By the same way, we get ϕ|dP |Z22  C(n, k)μk H22

√

16

εϕP 22

(6.5)

Now, by Lemma 4.2, we have (|H| −

H2 )ϕP 22

mk 2  P Sn vM

 ||H| − H2 |2 |X|2k ϕ2 dv M

C(n, k) 2 P Sn ϕ(|H| − H2 )22 H2k 2 √  C(n, k)μk H22 16 εϕP 22 

(6.6)

By the same way, we get ϕ(|H| − H2 )dP (ν)22  C(n, k)μk

√

16

εH42 ϕP 22

(6.7)

Now let us estimate the last terms of (6.1) ϕ|∇0 dP ||Z|22 

C(n, k)μk 2 P Sn vM

 ϕ2 |X|2k−4 |Z|2 dv M



C(n, k)μk H42

√

16

εϕP 22

(6.8)

Reporting (6.3), (6.4), (6.5), (6.6), (6.7) and (6.8) in (6.1) we get √ Δ(ϕP ) − μk H22 ϕP 2  C(n, k) 16 εμk H22 ϕP 2 .

2

Let Ekε be the space spanned by the eigenfunctions of M associated to an eigenvalue $ % √ √ in the interval (1 − 16 ε2C(n, k))μSk M , (1 + 16 ε2C(n, k))μSk M . If dim Ekε < mk , then  there exists ϕP ∈ (ϕHk (M )) \ {0} which is L2 -orthogonal to Ekν . Let ϕP = fi be the i

decomposition of ϕP in the Hilbert basis given by the eigenfunctions fi of M associated respectively to λi . Putting N := {i/ fi ∈ / Ekε }, by assumption on P we have  2 √ λi − μSk M fi 22 = Δ(ϕP ) − μSk M ϕP 22 4C(n, k)2 8 ε(μSk M )2 ϕP 22  i∈N

√  (μSk M )2 C(n, k)2 8 εϕP 22 which gives a contradiction. We then have dim Ekε  mk . This complete the proof of Theorem 1.1.

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7. Proof of Theorem 2.1 We adapt the constructions made in [1,10,4]. There will be two steps. We first consider submanifolds obtained by connected sum of a small submanifold εM2 with a fixed submanifold M1 along a small, adequately pinched cylinder εTε . Note that contrary to the constructions in [1,10], this is a 2 scales collapsing sequence of submanifolds. It will first give Theorem 2.1 in the case where F \ Sp(M1 ) is a singleton. We will then get the general case by iterating the construction (i.e. by glueing several such cylinders). Subsequently, for any subset A of Rn+1 , we denote by λA the set obtained by applying an homothety of factor λ to A. 7.1. Case F = Sp(M1 ) ∪ {λ} 7.1.1. Flattening of submanifolds ˜ ε a submanifold For any submanifold M of Rn+1 and ε > 0 small enough, we set M n+1 of R obtained by smooth deformation of M at the neighborhood of a point x0 ∈ ˜ ε and M ˜ ε \ Bx (10ε) is a subset of M . We also set M such that Bx0 (4ε) is flat in M 0 ε ε ˜ M = M \ Bx0 (3ε) whose boundary has a neighborhood isometric to the flat annulus ˜ ε in [4] B0 (4ε) \ B0 (3ε) in Rm . We describe precisely how to construct such flattening M so that it also satisfies the following curvature estimates for any α  1.  lim

ε→0 ˜ε M

ε→0 Mε

 lim

ε→0 ˜ε M



 |Hε |α dv = lim

|Hε |α dv = M



 |Bε |α dv = lim

ε→0 Mε

|H|α dv

|Bε |α dv =

|B|α dv M

˜ ε ) tends to isometric to H 1 (M ) as ε tends to 0. Note that H 1 (M For more convenience in this section the norms in the different spaces will not be normalized by the volume. 7.1.2. A small manifold with a prescribed eigenvalue Let M1 , M2 be 2 manifolds of dimension m isometrically immersed in Rn+1 and λ, L 2 be some positive real numbers with λ ∈ / Sp(M1 ) and L > max(1, C(M1 ,Dd12)(1+λ) ), where d is the distance λ to Sp(M1 ) and C is a constant that will be fixed later. ˜ η of M2 around the point x2 Let 0 < η < 1 small enough such that the flattening M 2 exists. Let D be a smooth hypersurface of revolution in Rm+1 , composed of three parts, D1 , D2 , D3 , where D1 is a cylinder of revolution isometric to B0 (3) \ B0 (2) ⊂ Rm+1 at the neighborhood of one of its boundary component and isometric to [0, 1] × Sm−1 at the neighborhood of its other boundary component, where D2 = [0, L] × Sm−1 and

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where D3 is a disc of revolution with pole x3 and isometric to [0, 1] × Sm−1 at its boundary and to a flat disc at the neighborhood of x3 . Let C be a cylinder of revolution of dimension m isometric to B0 (2) \ B0 (1) ⊂ Rm at the neighborhood of its 2 boundary components.

˜ η \Bx (2ν), of νC and of D\Bx (2ν) along For any ν < η/4 small enough, the gluing of M 2 3 2 their isometric boundary components exists and is a smoothly immersed submanifold Tν of dimension m. By now classical arguments (see for instance [1]), when ν tends to 0, the Dirichlet spectrum of Tν converges to the disjoint union of the Dirichlet spectrum of D and of ˜ η . In particular, the limit spectrum has 0 as isolated eigenvalue with the spectrum of M 2  multiplicity one. Moreover, since λD 1 (Tν ) has multiplicity one, it depends continuously on ν. ˜ η , L, D1 , D3 , C) there exists νε < ν0 (λ, M ˜ η , L, D1 , We infer that for any ε < ε0 (λ, M 2 2 η  2 D  ˜ D3 , C) such that λD 1 (Tνε ) = ε λ and λ2 (Tνε )  Λ2 (λ, M2 , L, D1 , D3 , C) > 0. We set Tε = εTν ε . Note that for any ε  ε0 , we have

D λD 1 (Tε ) = λ and λ2 (Tε ) 

Λ2 ε2

(7.1)

7.1.3. Gluing and control of its curvature Now let x1 ∈ M1 and ζ > 0 fixed. We first assume that Bx1 (4ζ) ∪ M1 is flat. For any ˜ 4ζ \ Bx (3ε). So we set Mε the m-submanifold of Rn+1 obtained ε < ζ, we set M1ε = M 1 1 ε by gluing M1 and Tε along their boundaries in a fixed direction ν ∈ Nx1 M1 . Note that Mε is a smooth immersion iε of M1 #M2 (resp. an embedding when M1 and M2 are embedded).

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

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By the computations above, the sequence iε (M1 #M2 ) = Mε converges to M1 in Hausdorff distance and we have 

⎛ ⎜ |Hε |α dv  εm−α ⎝

Mε



⎛ ⎜ |Bε |α dv  εm−α ⎝



 |Hε |α dv + C(m, α)L +

M2η ∪νε C∪D1 ∪D3

M1



 |Bε |α dv + C(m, α)L +

M2η ∪νε C∪D1 ∪D3

Mε

⎞ ⎟ |Hε |α dv ⎠

(7.2)

⎞ ⎟ |Bε |α dv ⎠ .

(7.3)

M1

7.1.4. Computation of the spectrum of Mε We will prove that there exists a sequence (εp )p∈N such that εp → 0 and the spectrum ˜ where λ ˜ satisfies λ − of Mεp converges to the disjoint union of Sp(M1 ) and of {λ}, C(M1 ,D1 )(1+λ) ˜  λ. The collapsing of Mε is multiscale, after rescaling of Tε , we √  λ L get another collapsing sequence of submanifolds with no uniform control of the trace and Sobolev Inequalities, so the cutting and rescaling technique of [1,10] does not work directly in our case and need to be adapted. We denote by (λk )k∈N the union with multiplicities of the spectrum of M1 and of {λ}, by (λεk )k∈N the spectrum of Mε and by (μεk )k∈N the Dirichlet spectrum of the disjoint union Tε ∪ M1ε . By the Dirichlet principle, we have λεk  μεk for any k ∈ N. It is well known (see for instance [6]) that the Dirichlet spectrum of M1ε converges to the spectrum of M1 . We infer that μεk → λk as ε → 0 and so lim sup λεk  λk for any k ∈ N. ε→0

We set αk = lim inf λεk . To get some lower bound on the αk , we need some local trace ε→0

inequalities at the neighborhood of ∂M1ε . Local trace inequalities. We set St = {x ∈ Tε | d(x, ∂Tε ) = −t} for any t  0 and St = {x ∈ M1ε | d(x, ∂M1ε ) = t} for any t  0. Obviously we have ∂Tε = S0 = ∂M1ε . Let εl be the distance in Mε between M1ε and εD2 (i.e. l is the distance between the two boundary components of D1 in D1 ).

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Let η : [−(1 + L + l)ε, ζ] → [0, 1] be a smooth function such that η(t) = 1 for any t  ζ2 , η(ζ) = 0 and |η  |  ζ4 . For any r ∈ [−(1 + L + l)ε, ζ] and any f ∈ H 1 (Mε ), we have: 

 f 2 dσr =

Sr

f (r, u)2 θε (r, u)du S0

 = S0

⎛ ζ ⎞2  ∂ ⎝ [η(·)f (·, u)]ds⎠ θε (r, u)du ∂s r

⎛ ⎞2   ζ  ( ∂ ds ⎠ θε (r, u)du = ⎝ [η(·)f (·, u)] θε (r, u) ( ∂s θε (r, u) S0

r

S0

r

⎛ ⎞⎛ ζ ⎞ 2  ζ   θ (r, u) ∂ ε  ⎝ [η(·)f (·, u)] θε (s, u)ds⎠ ⎝ ds⎠ du ∂s θε (s, u) r

For r ∈ [−ε, ζ] and m  3: ζ r

θε (r, u) ds = θε (s, u)

ζ 1+

1+ r

 r m−1 3ε ds  s m−1 3ε

ζ/(3ε)  ! dt r "m−1 = 3ε 1 + 3ε (1 + t)m−1 r/(3ε)

) *ζ/(3ε) ! r "m−1 3ε 1 = 1+ − 3ε m−2 (1 + t)m−2 r/(3ε)  C(m)(ε + |r|) And if r ∈ [−(1 + L + l)ε, −ε] using the fact that θε (s, u) is increasing in s we have: ζ r

θε (r, u) ds  θε (s, u)

−ε r

θε (r, u) ds + θε (s, u)

ζ

−ε

θε (−ε, u) ds  C(m)(ε + |r|) θε (s, u)

which gives  Sr

⎡⎛ ⎢ f 2 dσr  C(m)(ε + |r|) ⎣⎝

 ζ S0 r

⎞1/2 16 2 f (s, u)θε (s, u)dsdu⎠ ζ2

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

⎛ +⎝

 ζ 

2

∂ f (·, u) ∂s

1235

⎞1/2 ⎤2 ⎥ θε (s, u)dsdu⎠ ⎦

S0 r

 c(ζ)(ε + |r|)f 2H 1 (Mε ) .

(7.4)

First estimates on eigenfunctions. We now use this local trace inequality to get some estimates on the eigenfunctions

of Mε . We set ϕ : Mε → [0, 1] be a smooth function equal to 1 on M ε \ Ss , equal to 0 on on

√ −(l+L− L)εsε/2



Ss and such that |dϕ| 

4 ε

−(l+L)εsε



on

Ss and |dϕ| 

2 √ ε L

ε/2sε

Ss . For any f1 , f2 ∈ H (Mε ), integration of Inequality (7.4) 1

√ −(l+L)εs−(l+L− L)ε

gives us          f1 f2 dv −  ϕf ϕf dv |ϕ2 − 1||f1 ||f2 |dv 1 2     Mε

Mε

Mε

⎛

ε

⎝

 −(d+L)ε



⎞1/2 ⎛ |f1 |2 dσs ⎠



⎝

Ss

⎞1/2 |f2 |2 dσs ⎠

ds

Ss

ε  c(ζ)f1 H 1 (M ε ) f2 H 1 (M ε )

(ε + |s|)ds

−(l+L)ε

 c(ζ, l, L)ε f1 H 1 (Mε ) f2 H 1 (Mε ) 2

(7.5)

√ and putting Iε = [−(l + L)ε, −(l + L − L)ε] ∪ [ε/2, ε] we have   2 |d(ϕf1 )| dv  (|dϕ|2 f12 + 2ϕf1 (df1 , dϕ) + ϕ2 |df1 |2 )dv Mε

Mε

16  2 ε

 Iε

⎛ 8 + ⎝ ε

 Iε



⎛ ⎝

 Ss

⎛ ⎝



⎞ f12 dσs ⎠ ds ⎞

⎞1/2 ⎛

f12 dσs ⎠ ds⎠

Ss

c(ζ, l, L)f1 2H 1 (Mε )

⎝



Mε

⎞1/2 |df1 |2 dv ⎠

 +

|df1 |2 dv

Mε

(7.6)

Let (fkε ) be a L2 -orthonormal, complete set of eigenfunctions of Mε . For any k, we set f˜kε the function on M1 equal to ϕfkε on M1ε and extended by 0. By Inequality (7.6), we

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

1236

have f˜kε 2H 1 (M1 )  c(M1 )(1 + λk ) for ε small enough. We infer by diagonal extraction ε that there exists some sequences (εp )p∈N and (hk )k∈N ∈ H 1 (M1 )N such that λkp → αk ε and (f˜k p )p converges weakly in H 1 (M1 ) and strongly in L2 (M1 ) to hk , for any k. It is easy to prove that hk is a weak solution of Δhk = αk hk on H 1 (M1 \ {x1 }) = H 1 (M1 ) (see for instance [10]). By elliptic regularity, either hk = 0 or αk is an eigenvalue of M1 . Estimate (7.4) will not be good enough to control the eigenvalues λεk whose eigenfunctions tends to concentrate on Tε so we need to improve it. Improved estimate on eigenfunctions. ε Let k0 ∈ N such that λk0 = λ. Since D2 isometric to [0, L] × Sm−1 , any fk p can be  ε ε seen as a function on [0, εp L] × εp Sm−1 . For any f = βi fi p ∈ Vect{fi p | i  k0 }, ik0

m

we define the rescaling Fp on c = [0, 1] × Sm−1 by Fp (t, x) = εp2 Inequality (7.4), we have  Fp2 dv = c

1 ε2p L2

 f 2 dv = εp D2

⎛

c(M1 ) ⎜  2 2 ⎝ εp L  = c(ζ)  Fp2 dv {0}×Sm−1

1 = Lεp 

1 ε2p L2

⎛

−εl

⎝

−ε(L+l)

L− 2 f (εp Lt, εp x). By 1

⎞



f 2 dσr ⎠ dr

Sr

⎞

−εl

−1

⎟ (ε + |r|)dr⎠ f 2H 1 (Mε )

−ε(L+l)

l+1 1 + L 2





(1 + λ)f 22 

1 f dv = Lεp 2

εp (D1 ∩D2 )

f 2 dσ−dεp S−dεp

c(ζ) c(M1 , D1 ) (1 + l)(1 + λ)f 22 = (1 + λ)f 22 L L

(7.7)

and  Fp2 {1}×Sm−1

1 = Lεp





1 f = Lεp 2

εp (D3 ∩D2 )



 c(ζ) 1 +

1+l L

f 2 dσ−(l+L)εp S−(l+L)εp



(1 + λ)f 22

(7.8)

Moreover, we have 

 |dFp |2 dv = c

[0,1]×Sm−1



 |dFp |2 dtdx =

εm p L [0,1]×Sm−1

∂f ∂s

2 (εp Lt, εp x)dtdx

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242



−1 εp S εm |d p L

+

m−1

1237

f |2 (εp Lt, εp x)dtdx

[0,1]×Sm−1



 =

∂f ∂s

2



1 dv + 2 L

εp D2

|dεp S

m−1

f |2 dv

εp D2



|df |2 dv  λf 22

 Mε

Note that we have used the fact that L > 1. So we can assume that there exists F∞ ∈ 1 2 H 1 (c) such that the  sequence (Fp ) converges to F∞ weakly in H (c) and strongly in L (c). We set jp (t) =

F∞ (t, x)dx, we have jp , j∞ ∈ H 1 ([0, 1])

Fp (t, x)dx and j∞ (t) = Sm−1

Sm−1

∂Fp (t, x)dx), jp → j∞ strongly in L2 ([0, 1]) and weakly in H 1 ([0, 1]). ∂t

(with jp (t) =

Sm−1

By the estimates (7.7) and (7.8) and the compactness of the trace operator on c, we have (

|j∞ (0)| 

c(ζ)

(1 + l)(1 + λ)f 2 √ L

and 1 |j∞ (1)|  c(ζ)

1+

1 + l√ 1 + λf 2 L

 Hence (t) = j∞ (t) − j∞ (0) + (j∞ (1) − j∞ (0))t is in H01 ([0, 1]). For any ψ ∈ Cc∞ ([0, 1]), we set ψp (t, x) = εp Lψ( εptL ) seen as a function in H01 (εp D2 ). We have 1



1



 ψ dt = 0

 j∞ ψ  dt

0

1 = lim p

jp (t)ψ  (t) dt = lim p

c

0

= lim p





ε βi λi p m

i

εp2 m

√

L

−1



ε

∂Fp  1 ψ dv = lim m √ p 2 ∂t εp L

fi p ψp dt dx =

 i

εp D2 ε

 df, dψp dt dx εp D2

 αi βi L2 lim ε2p

Fi,p ψ dt dx = 0,

p

c

where Fi,p (t, x) = εp2 L− 2 fi p (εp Lt, εp x). We infer  is harmonic and in H01 ([0, 1]), i.e.  = 0 and j∞ (t) = j∞ (0) + (j∞ (1) − j∞ (0))t on [0, 1]. Since the Poincare inequality on Sm−1 gives us 1

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

1238

⎛

 Fp (t, x)2 dx  Sm−1



1 ⎝ Vol Sm−1

⎞2



Fp (t, x) dx⎠ +

Sm−1

1 εp j 2 (t) + Vol Sm−1 p (m − 1)L

1 m−1



 |dSm−1 Fp |2 dx Sm−1

|dεp Sm−1 f |2 (εp Lt, x) dx, εp Sm−1

we get that 1 Lε2p



 2

Fp2 dtdx

f dv = L √ [0,εp L]×εp Sm−1

[0, √1L ]×Sm−1

√1

L  Vol Sm−1

L jp2 (t) dt 0



1 + (m − 1)L [0,εp

√ L]×εp Sm−1

|dεp Sm−1 f |2 dv

√1

L  Vol Sm−1

L jp2 (t) dt + 0

λ f 22 (m − 1)L

√1

→

L Vol Sm−1

L j∞ (t)2 dt + 0

λ f 22 (m − 1)L √1

L j∞ (t)2 dt 

Now a straightforward computation shows that

c(ζ)(1 + l)(1 + λ)f 22 L3/2

0

and for p great enough 1 Lε2p

 f 2 dv  √ [0,εp L]×εp Sm−1

c(ζ)(1 + λ)(1 + l)f 22 √ L

(7.9)

Note that this estimate is better than which could be deduced from (7.4). Control of the limit spectra. If the family (hi )i
i=0

i=0

αi = μ for any i such that βi = 0 (we recall that λkp → αk for any k ∈ N and λk0 = λ). k 0 −1 ε Setting uεp = βi fi p we then have i=0

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

1239

                  |d(ϕuεp )|2 dv − μ (ϕuεp )2 dv  =  (|dϕ|2 u2εp + ϕ2 uεp Δuεp )dv − μ (ϕuεp )2 dv       Tεp  Tεp  Tεp Tεp    4 ε ε ε  u2εp dv + (λi p − μ)βi ϕfi p βj ϕfj p dv Lε2p √ i,j Tεp

εp ([−(l+L),−(l+L− L)]×Sm−1 )

√ √ Since εp ([−(l + L), −(l + L − L)] × Sm−1 ) is isometric to [0, εp L] × εp Sm−1 we deduce from (7.9) that          2 2 |d(ϕuεp )| dv − μ (ϕuεp ) dv     Tεp  Tεp   c(ζ)(1 + l)(1 + λ) ε ε ε √  (λi p − μ)βi ϕfi p βj ϕfj p dv (7.10) uεp 22 + L i,j Tεp

ε

We recall that ϕfk p |M converges strongly in L2 (M1 ) and then ϕuεp converges strongly 1      k −1 0     in L2 (M1 ) to βi hi = 0. Moreover from (7.5) we have  (ϕuεp )2 dv − 1 → 0. Then   i=0 Mεp   we deduce that (ϕuεp )2 dv → 1. Since ϕuεp |T ∈ H01 (Tεp ) and since by construction εp

Tεp



 |d(ϕuεp )|2 dv  λ

of Tεp , we have λD 1 (Tεp ) = λ = λk0 , we then have Tεp

(ϕuεp )2 dv.

Tεp

Then for p large enough  0 < (λ − λk0 −1 )

 (ϕuεp )2 dv  (λ − μ)

Tεp



(ϕuεp )2 dv

Tεp



|d(ϕuεp )| dv − μ



2

Tεp

(ϕuεp )2 dv

Tεp

( From now we assume that C(M1 , D1 ) > c(ζ)(1 + l). Letting p tend to ∞ in (7.10) we )(1+λ) √1 get that d  λ − λk0 −1  C(M1 ,D which contradicts the choice made on L. L We infer that (hi )i
E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

1240

Assume that thereexists another index k1 = k0 such that hk1 =0. Then, Inequalε ε ε ε ity (7.5) gives that ϕfk0p ϕfk1p dv → 0, (ϕfk0p )2 dv → 1 and (ϕfk1p )2 dv → 1 Tεp

Tεp



ε

|d(ϕfk0p )|2 dv and

and Inequality (7.6) gives that

Tεp



Tεp

ε

|d(ϕfk1p )|2 dv remain bounded as Tεp

εp → 0. We set gp a unitary eigenfunction of Tεp for the Dirichlet problem associated to ε ε the eigenvalue λ. If we set (ϕfk0p )|Tεp = βkp0 gp + γkp0 and (ϕfk1p )|Tεp = βkp1 gp + γkp1 , with p p p p 1 βk0 , βk1 ∈ R and γk0 , γk1 orthogonal to gp in H0 (Tεp ). The previous relations and the lower bound on λD 2 (Tεp ) imply that 

ε

p 2 p 2 |d(ϕfk0p )|2 dv  λ(βkp0 )2 + λD 2 (Tεp )γk0 L2 (Tεp )  (βk0 ) λ + Tεp

By the same way, (βkp1 )2 λ +

p 2 Λ2 ε2p γk1 L2 (Tεp )

Λ2 p 2 γ  2 . ε2p k0 L (Tεp )

is bounded, and so γkp0 2L2 (Tε

and p) p 2 p 2 p tend to 0 with εp . Now, we have (βk0 ) + γk0 L2 (Tε ) → 1 and so |βk0 | → 1. p ε Up to change of sign of fk0p , we can assume that βkp0 → 1. By the same way, we have  ε ε |βkp1 | → 1, which contradicts the fact that ϕfk0p ϕfk1p dv → 0. We infer that for any γkp1 2L2 (Tε ) p

Tεp

k ∈ N \ {k0 } we have that αk is an eigenvalue of M1 . Moreover, if we decompose ε (ϕfk p )|Tεp = βkp gp + γkp as above, Inequality (7.6) implies that (βkp )2 + Λε22 γkp 2L2 (Tε ) p

p

remains bounded and so we have lim γkp 2L2 (Tεp ) = 0 and Inequality (7.5) gives p→∞

 0 = lim

p→∞ Mε

ε

ε

fk0p fk p dv = lim βkp βkp0 = lim βkp p→∞

ε

and so (ϕfk p )|Tεp → 0 in L2 (Tεp ) for any k = k0 . Once again, Inequality (7.5) gives us that for any k, k ∈ N \ {k0 }, we have 

 hk hk dv = lim

M1

p→∞ M1

 = lim

p→∞ Mεp

ε



ε

ϕfk0p ϕfkp dv = lim

ε

ε

ϕfk0p ϕfkp dv

p→∞ ε

M1 p ε

ε

ϕfk0p ϕfkp dv = δkk

From the min–max principle, it gives that we have αk  λk for any k = k0 . Since we have αk  λk for any k ∈ N, we infer that for any k ∈ N \ {k0 } we have αk = λk . Moreover ε we have αk0  λk0 = λ. Finally, since ϕfk0p | ∈ H01 (Tεp ), Inequality (7.10), applied to ε

f = fk0p and μ = αk0 gives that

Tεp

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

 (λ − αk0 )



ε

(ϕfk0p )2 dv 

Tεp



ε

|d(ϕfk0p )|2 dv − αk0 Tεp



1241

ε

(ϕfk0p )2 dv Tεp

c(ζ)(1 + l)(1 + λ) √ + L



ε

ε

(λkp0 − αk0 )(ϕfk0p )2 dv

Tεp

ε Now we have seen that f˜k0p tends to hk0 = 0 in L2 (M1 ). It follows that from (7.5), ) *  c(ζ)(1 + λ)(1 + l) ε √ lim (ϕfk0p )2 dv = 1 and we deduce that αk0 ∈ λ − ,λ . p→∞ L Tεp

At this stage of the proof, we get that for any sequence (εk ) such that lim εk = 0, k

the sequence (Mεk )k∈N of immersions of M1 #M2 satisfies the point (1), (2) * and (4) of ) c(ζ)(1 + λ)(1 + l) √ Theorem 2.1 and we have lim Sp(Mεk ) ⊂ Sp(M1 ) ∪ λ − ,λ . k L By an  extraction taking  exists a subsequence (εp(i) ), such  easy diagonal  L = i, there α α α that |H| dv → |B| dv and |B| dv → |B|α dv for any α < m (see (7.2) Mεp(i)

M1

Mεp(i)

M1

and (7.3)) and we get Theorem 2.1 for F = Sp(M1 ) ∪ {λ} when Bx1 (4ζ) is flat in M1 . ˜ ζ ) converges to Now if we assume that Bx1 (4ζ) is not flat, we use the fact that Sp(M 1 Sp(M1 ) and by a new diagonal extraction we get the desired result. 7.2. End of the proof of Theorem 2.1 Let F be a closed subset containing Sp(M1 ). As explained in section 2, there exists an increasing sequence of finite sets FN such that FN ⊂ [0, N ] ∩F ⊂ FN,1/N . We can assume that Sp(M1 ) ∩ [0, N ] is contained in FN . Thus FN = GN ∪ (Sp(M1 ) ∩ [0, N ]) where GN and Sp(M1 ) ∩ [0, N ] are disjoint and GN is finite. First we have F = LimSetFN , FN N →∞

converges to F for the distance of Attouch–Wetts–Hausdorff as well as GN ∪ Sp(M1 ) converges to F . Now, iterating the construction (with M2η replaced by Sm for any supplementary gluing) we obtain a sequence MN,εp such that Sp(MN,εp ) converges to GN ∪ Sp(M1 ) when p tends to infinity. Since GN ∪Sp(M1 ) converges to F when N tends to infinity, by diagonal extraction there exists subsequences (Nk )k and (εk )k such that LimSetSp(MNk ,εk ) = F k→∞

and the point (2) of the Theorem  2.1 on the curvatures is true. In the case α = m, the limit |B|m dv depends on L and so we are only able to get a Mε

weak version of Theorem 2.1 with F = Sp(M1 ) ∪G, where G is a finite set whose elements  |B|m dv

are known up to an error term and where the point (2) is replaced by ik (M1 #M2 )

is bounded by a constant that depend on M1 , M2 , D1 , D3 , G and on the error term.

1242

E. Aubry, J.-F. Grosjean / Journal of Functional Analysis 271 (2016) 1213–1242

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