Entire radially symmetric graphs with prescribed mean curvature in warped product spaces

J. Math. Anal. Appl. 441 (2016) 393–402

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Entire radially symmetric graphs with prescribed mean curvature in warped product spaces Zonglao Zhang College of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang, 325035, PR China

a r t i c l e

i n f o

Article history: Received 1 September 2015 Available online 5 April 2016 Submitted by H.R. Parks Keywords: Graph Hyperbolic space Mean curvature Riemannian manifold Warped product

a b s t r a c t This study investigates entire radially symmetric graphs in the warped product M ×f R, where M is a Riemannian manifold with a pole and f : M → R+ is the warping function. The main results demonstrate the existence and uniqueness of entire radially symmetric graphs with prescribed mean curvature. © 2016 Elsevier Inc. All rights reserved.

1. Introduction The study of entire graphs with certain curvature properties plays an important role in classical differential geometry. In 1915, Bernstein published his well-known result that an entire minimal graph in three-dimensional Euclidean space R3 must be a plane (see [12]). In 1955, Heinz proved that if a graph in 1 R3 defined by z = ϕ(x, y), x2 + y 2 < R2 , has constant mean curvature H, then |H| ≤ R (see [8]). This implies that the graph must be minimal if it is entire, and it should be a plane according to Bernstein’s result. Subsequently, this topic was addressed by many mathematicians and it is still a very active research area. In the last few decades, many authors have extended the study of graphs from graphs in Euclidean space to those in hyperbolic space (e.g., see [9,10,13]), and even to those in warped product spaces (e.g., see [1–3,6,14,15]). This study investigates entire radially symmetric graphs in the warped product M = M ×f R, where M is a Riemannian manifold, R is the 1-dimensional Euclidean space with the standard metric, and f is a positive smooth function on M . We obtain some existence and uniqueness results for entire radially symmetric graphs with prescribed mean curvature. In the main results, the Riemannian manifold M is a radially symmetric manifold with a pole, or M is the n-dimensional hyperbolic space Hn . E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.jmaa.2016.03.088 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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The remainder of this paper is organized as follows. In Section 2, we recall some notations and known results. In Section 3, we give our main results and their proofs. 2. Preliminaries First, we recall the notation for the warped product (see [5,11]). Let (M, g) and (N, g˜) be Riemannian manifolds with metrics g and g˜, respectively, and f is a positive smooth function on M . Let π and π ˜ be the projections of M × N onto M and N , respectively. The warped product M ≡ M ×f N is the product manifold M × N equipped with the warped product metric g¯ = π ∗ (g) + (f ◦ π)2 π ˜ ∗ (˜ g ). We refer to f as the warping function of the warped product. If the warping function f = 1, then the warped product M ×f N is simply the usual Riemannian product M × N . In this study, we only consider the warped products M = M ×f R, especially the case where M = Hn . In this section and in the sequel, Hn denotes the n-dimensional hyperbolic space with sectional curvature −1 and R denotes the Euclidean line. It is interesting that Hn ×f R = Hn+1 when f (x) = cosh r(x) for any x ∈ Hn , where r(x) is the hyperbolic distance from x to some fixed point in Hn (see [10,16]). We refer to Hn ×cosh r R as the warped product model of Hn+1 . Let M = M ×f R, ϕ ∈ C ∞ (Ω), where Ω is an open subset of M . We call Σ = {(x, ϕ(x)) | x ∈ Ω} ⊂ M the graph of the function ϕ in the warped product M ×f R. When Ω = M , the graph Σ is said to be entire. The graph Σ of the function ϕ is a smooth submanifold of M . Let ∇ and Δ be the gradient and Laplacian of M , respectively. The mean curvature H of the graph Σ (with respect to the upward pointing normal vector of Σ) is given by (see [14,15]) nH = ρΔϕ + (∇ϕ)ρ +

ρ (∇ϕ)f, f

(1)

where n = dim M and ρ= 

1 1 f2

+ ∇ϕ2M

,

(2)

where ∇ϕM denotes the norm of the vector ∇ϕ in the metric of M . Next, we recall the notation of a manifold with a pole (see [7]). Let M be a Riemannian manifold and o ∈ M . The point o is said to be a pole of M if the exponential map expo : To M → M is a diffeomorphism, where To M denotes the tangent space of M at o. If M is a manifold with a pole o, we usually write (M, o) for M to clearly denote the pole under consideration. Let (M, o) be a manifold with a pole. M is said to be radially symmetric around o if for any α, α ˜ ∈ To M such that |α| = |˜ α|, an isometry Ψ : M → M exists such that Ψ(o) = o and Ψ∗ |o (α) = α, ˜ where |α| denotes the length of vector α and Ψ∗ |o denotes the differential of Ψ at o. It should be noted that [7] used the term “weak model” to refer to our “radially symmetric manifold.” Let M be radially symmetric around o. For any x ∈ M , let r(x) ≡ dist(o, x), i.e., the geodesic distance from o to x. If φ(t) is a C ∞ function defined on an interval I ⊂ [0, +∞), and if we let ϕ(x) ≡ φ(r(x)), then ϕ(x) defines a function on Ω = {x|r(x) ∈ I, x ∈ M }. We call ϕ(x) = φ(r(x)) a radially symmetric function around o. For the sake of convenience, we often write ϕ(x) = ϕ(r(x)) for the radially symmetric

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function ϕ(x) = φ(r(x)). In the warped product space M ×f R, the graph of a radially symmetric function (around o) is called a radially symmetric graph (around o). If ϕ = ϕ (r(x)) is a smooth radially symmetric function, then we have Δϕ = ϕ + (Δr)ϕ ,

(3) 2

d ϕ  where all the derivatives are with respect to r, i.e., ϕ = dϕ dr , ϕ = dr 2 . It is obvious that Δϕ is radially symmetric if ϕ is a smooth radially symmetric function. For the radially symmetric graph Σ of a radially symmetric function ϕ around o in the warped product M ×f R, if the warping function f is also radially symmetric around o, then by a straightforward computation (see [14]), we can rewrite the mean curvature equation (1) of Σ as

ρ   ϕf , f

nH = ρ [ϕ + (Δr)ϕ ] + φ ρ +

(4)

where n = dim M . In this case, by (2), we have ρ= 

1 1 f2

+ ∇ϕ2M

=

f 1 + (ϕ f )2

(5)

and thus ρ =

f  − f 3 ϕ ϕ 3 . [1 + (ϕ f )2 ] 2

(6)

Thus, by applying (5) and (6) to (4), we obtain f (ϕ + ϕ Δr) nH =  + ϕ 1 + (ϕ f )2



f  − f 3 ϕ ϕ 3 [1 + (ϕ f )2 ] 2



ϕ f  + . 1 + (ϕ f )2

(7)

3. Main results and their proofs In this section, we study entire graphs in the warped product M ×f R with prescribed mean curvature. Our aim is to solve the following natural problem. Let M be a Riemannian manifold with a pole o. Suppose that M is radially symmetric around o. Consider the warped product M ×f R such that f is radially symmetric around o. For a given smooth  radially symmetric function H(r) on M , can we find an entire radially symmetric graph Σ such that  the mean curvature of Σ is H(r)? If such an entire graph exists, is it unique? To answer this problem, we need to solve the second order nonlinear differential equation (7). A nonlinear differential equation is usually difficult to solve. However, in the following lemma, we find that Equation (7) can be changed into a first-order linear differential equation by introducing a transformation. Lemma 3.1. Let (M, o) be an n-dimensional Riemannian manifold with a pole. Consider the warped product M ×f R. Let ϕ be a C ∞ function on M and Σ be the graph of ϕ. Denote H as the mean curvature of Σ. Suppose that M , f , and ϕ are radially symmetric around o. Set Φ= 

ϕ f 1 + (ϕ f )2

.

(8)

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Then, Φ satisfies the first-order linear differential equation Φ +



 f + Δr Φ = nH f

(9)

for r > 0, where r = r(x) is the geodesic distance from o to x ∈ M (all of the derivatives in the lemma are with respect to r). Proof. From (8), we have

Φ = =

(ϕ f )

  f )(ϕ f )  1 + (ϕ f )2 − (ϕ f ) (ϕ 1+(ϕ f )2 1 + (ϕ f )2

(ϕ f ) 3 . [1 + (ϕ f )2 ] 2

(10)

Using (7) in conjunction with (8) and (10), we obtain    ϕ f  f [ϕ + ϕ Δr] f − f 3 ϕ ϕ + nH =  + ϕ 3 [1 + (ϕ f )2 ] 2 1 + (ϕ f )2 1 + (ϕ f )2   f  − f 3 ϕ ϕ f f ϕ Φ + (Δr)Φ + = + ϕ 3 f [1 + (ϕ f )2 ] 2 1 + (ϕ f )2

 f ϕ + ϕ f  f + Δr Φ = + 3 f [1 + (ϕ f )2 ] 2

 f = Φ + + Δr Φ. f Thus, (9) holds and the lemma is proved. 2 Next, we provide a theorem regarding the existence and uniqueness of entire graphs with prescribed mean curvature in the warped product space M = M ×f R, where M is a Riemannian manifold with a pole o, M is radially symmetric around o, and f = f (r(x)) is a smooth positive radially symmetric function on M . Before stating the theorem, we introduce some notations and provide some preliminary discussion. Denote Sr as the geodesic sphere of M with center o and radius r, and A(r) as the volume of Sr . Set h(r) =  r 0

f (r)A(r) f (ζ)A(ζ)dζ

(11)

for r > 0. It is known (see [17, Lemma 1]) that lim rΔr = n − 1,

r→0

where n = dim M . (We remark that in [17], the above fact is specified for a so-called strongly symmetric manifold, but we can prove that it is still true for our radially symmetric manifold by a similar argument to that given in [17, Lemma 1].) Then, we have lim Δr = +∞.

r→0

In addition, by using L’Hôpital’s rule, we obtain

(12)

Z. Zhang / J. Math. Anal. Appl. 441 (2016) 393–402

lim h(r) = lim  r

r→0

r→0

0

397

f (r)A(r) A (r) A (r) = f (0) lim = lim . r→0 f (r)A(r) r→0 A(r) f (ζ)A(ζ)dζ

Thus, by employing the known fact (see [7, p. 44]) that Δr =

A , A

(13)

we obtain A (r) = lim Δr = +∞. r→0 A(r) r→0

lim h(r) = lim

r→0

(14)

If h(r) is strictly decreasing on (0, +∞), set λ = inf 0
(15)

r→+∞

Theorem 3.2. Let (M, o) be an n-dimensional Riemannian manifold with a pole, which is radially symmetric around o. Consider the warped product M = M ×f R, where f = f (r(x)) is a smooth positive radially   symmetric function on M , r(x) = dist(o, x). Let h(r) be the function defined by (11) and H(r) ≡ H(r(x)) is a given smooth radially symmetric function defined on the whole of M . Set λ = inf 0
λ , n

(16)

for all x ∈ M . Then, for a given constant ϕ0 ∈ (−∞, +∞), there exists a unique entire radially symmetric graph (around o) Σ = {(x, ϕ(r(x))) | x ∈ M }  in M such that: (i) ϕ|r=0 = ϕ0 ; (ii) the mean curvature of the graph is H(r(x)). Proof. The proof of the theorem involves constructing a smooth radially symmetric function ϕ = ϕ(r(x)) that satisfies conditions (i) and (ii) in the theorem, and proving the uniqueness of the function. Suppose that ϕ = ϕ(r(x)) is a smooth radially symmetric function around o defined on M such that the  mean curvature of the graph Σ of ϕ is H(r). By Lemma 3.1 and setting Φ= 

ϕ f 1 + (ϕ f )2

,

(17)

then Φ satisfies the first-order linear differential equation:



Φ +

f  + Δr Φ = nH. f

(18)

It should be noted that Equation (18) is singular at r = 0, so we first consider it on (0, +∞). Set

Φ0 (r) =

n f (r)A(r)

r 0

 H(ζ)f (ζ)A(ζ)dζ

r ∈ (0, +∞).

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Then, by using (13), we have ⎤ ⎡ r n  Adζ ⎦  A)2 − (f A) Hf ⎣H(f Φ0 = (f A)2 0

 = nH  = nH  = nH

(f A) Φ0 (r) − fA 

f A − + Φ0 (r) f A

 f + Δr Φ0 (r). − f

Thus, Φ0 (r) is a particular solution of Equation (18). It is also easy to verify that Φ1 (r) =

C f (r)A(r)

r ∈ (0, +∞)

(C is an arbitrary constant) is a general solution of the homogeneous equation 

Φ +



f + Δr Φ = 0. f

Therefore, from the general theory of first-order linear ordinary differential equations (e.g., see [4]), the  of Equation (18) on (0, +∞) is of the following form general solution Φ  Φ(r) = Φ1 (r) + Φ0 (r) C n + = f (r)A(r) f (r)A(r)

r

 H(ζ)f (ζ)A(ζ)dζ.

(19)

0

It should be noted that every solution of Equation (18) on (0, +∞) is among the general solution (19) and so is the function Φ(r) defined by (17). ϕ is radially symmetric around o and smooth at o, so we have ϕ |r=0 = 0. We then obtain Φ|r=0 = 0. By using L’Hôpital’s rule and (14), we note that n lim Φ0 (r) = lim r→0 r→0 f (r)A(r)

r

 H(ζ)f (ζ)A(ζ)dζ

0

 n H(r)f (r)A(r) lim =  r→0 f (0) A (r)  = nH(0) lim

r→0

A(r) A (r)

= 0. Thus, for Φ defined by (17), we must have C = 0 in (19), and therefore

Φ(r) = Φ0 (r) =

n f (r)A(r)

r 0

 H(ζ)f (ζ)A(ζ)dζ

(20)

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for r > 0 and Φ(0) = 0. By (17), we have (ϕ f )2 =

Φ2 . 1 − Φ2

By (20), we then obtain Φ2 − Φ2 )

(ϕ )2 =

f 2 (1

2   r  (ζ)A(ζ)dζ n 0 H(ζ)f  = 2    r  (ζ)A(ζ)dζ f (r)2 [f (r)A(r)]2 − n 0 H(ζ)f  for r > 0. By the hypotheses that h(r) is strictly decreasing on (0, +∞) and |H(r)| ≤ have h(r) > λ and

(21)

λ n,

then for r > 0, we

r f (r)A(r) > λ

f (ζ)A(ζ)dζ 0

r ≥n 0

  r        |H(ζ)|f (ζ)A(ζ)dζ ≥ n  H(ζ)f (ζ)A(ζ)dζ  .   0

Therefore, ⎡ ⎣n

r

⎤2  H(ζ)f (ζ)A(ζ)dζ ⎦ < [f (r)A(r)]2

(22)

0

for r > 0. Thus, from (21) and (22), we have    r   n  0 H(ζ)f (ζ)A(ζ)dζ    ϕ (r) = ±   2 r  f (r) [f (r)A(r)]2 − n 0 H(ζ)f (ζ)A(ζ)dζ for r > 0. By (17) and (20), for each r > 0, ϕ (r), Φ(r) and from (23), we obtain 

ϕ (r) =

r 0

 H(ζ)f (ζ)A(ζ)dζ have the same sign. Thus,

r

 H(ζ)f (ζ)A(ζ)dζ   2 r  f (r) [f (r)A(r)]2 − n 0 H(ζ)f (ζ)A(ζ)dζ n



(23)

0

(24)

for r > 0 and ϕ (0) = 0. It should be noted that the right-hand side of (24) is continuous and differentiable at r = 0 if we consider that it is 0 at r = 0. Thus, we obtain r ϕ(r) = 0

η

 H(ζ)f (ζ)A(ζ)dζ   2 dη + C1 , η  f (η) [f (η)A(η)]2 − n 0 H(ζ)f (ζ)A(ζ)dζ 

n

0

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where C1 is a constant. We use the initial condition that ϕ|r=0 = ϕ0 , so C1 = ϕ0 . Then, we have r ϕ(r) = 0

η

 H(ζ)f (ζ)A(ζ)dζ   2 dη + ϕ0 . η  f (η) [f (η)A(η)]2 − n 0 H(ζ)f (ζ)A(ζ)dζ 

n

0

(25)

This gives the uniqueness of the entire radially symmetric graph that satisfies conditions (i) and (ii) in the theorem. In addition, it is easy to verify that the function ϕ(r) defined by (25) satisfies ϕ |r=0 = 0 and that it defines a smooth radially symmetric function ϕ (r(x)) on M such that the graph of ϕ satisfies conditions (i) and (ii) in the theorem. This demonstrates the existence of the radially symmetric graph. Therefore, the proof of the theorem is complete. 2 Corollary 3.3. Using the same notations and assumptions employed in Theorem 3.2, we have the following. (1) Let H0 be a constant, then H0 is the mean curvature of an entire radially symmetric graph if and only if − nλ ≤ H0 ≤ nλ . (2) If λ ≡ inf 0 nλ , then by the hypothesis that h(r) is strictly decreasing on (0, +∞) and by (15), a constant r0 > 0 exists such that h(r) −λ < n|H0 | −λ when r > r0 . Then, for r > r0 , we have h(r) < n|H0 |, and hence  r    r   f (r)A(r) < n|H0 | f (ζ)A(ζ)dζ = n  H0 f (ζ)A(ζ)dζ  .   0

0

Therefore, ⎡ [f (r)A(r)]2 − ⎣n

r

⎤2 H0 f (ζ)A(ζ)dζ ⎦ < 0

0

for r > r0 . By (21), this means that (ϕ )2 < 0 for r > r0 , which is a contradiction. (2) If λ = 0, then by applying the first part of this corollary, the mean curvature H0 = 0, and by using (25), ϕ = constant. 2 Theorem 3.4. Let Hn+1 be the (n + 1)-dimensional hyperbolic space with sectional curvature −1. Consider the warped product model of Hn+1 : Hn+1 ≡ Hn ×cosh r R, where r = r(x) denotes the hyperbolic distance   from x ∈ Hn to a fixed point o ∈ Hn . Let H(r) ≡ H(r(x)) be a smooth radially symmetric function defined n  on the whole of H such that −1 ≤ H(r) ≤ 1. Then, an entire radially symmetric graph (around o) Σ = {(x, ϕ(r(x))) | x ∈ Hn }  in Hn ×cosh r R exists such that the mean curvature of the graph is H(r). The function ϕ(r(x)) is unique up to a constant.

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Proof. We apply Theorem 3.2 for M = Hn , f = cosh r. It is obvious that (Hn , o) is a manifold with a pole and that it is radially symmetric. In this case, we have (see [7, p. 52]) A(r) = σ(n)(sinh r)n−1 , where σ(n) = 2π n/2 /Γ( n2 ) and Γ denotes the gamma function. Now, the function h(r) defined by (11) has the expression h(r) =  r 0

f (r)A(r) f (ζ)A(ζ)dζ

(cosh r)(sinh r)n−1 (cosh ζ)(sinh ζ)n−1 dζ 0

= r

= n coth r for r > 0. It is easy to see that h(r) is strictly decreasing on (0, +∞) and λ=

inf

0
h(r) = n.

Thus, the hypothesis in Theorem 3.2 is satisfied, so Theorem 3.4 is an immediate result of Theorem 3.2. 2 Remark 3.5. In Theorem 3.4, we note the following. (1) If the mean curvature H0 = 0, then the entire radially symmetric minimal graphs are trivial slices ϕ = ϕ0 . If H0 = 1, then the entire radially symmetric graphs with mean curvature 1 are defined by ϕ = ln cosh r + C

(C is a constant),

which are simply the horospheres in the (n + 1)-dimensional hyperbolic space Hn+1 . (2) If the mean curvature of an entire radially symmetric graph is a constant H0 , then we must have |H0 | ≤ 1, as noted in Corollary 3.3. However, there is an entire radially symmetric graph with mean curvature H(r) that satisfies H(r) > 1 for all r = 0 (see [13, Lemma 6]). Acknowledgments The author would like to thank the referee for detailed comments and several helpful suggestions. This study was supported by the Zhejiang Provincial Natural Science Foundation of China (Project No. LY13A010009). References [1] A.L. Albujer, F.E.C. Camargo, H.F. de Lima, Complete spacelike hypersurfaces with constant mean curvature in −R ×Hn , J. Math. Anal. Appl. 368 (2010) 650–657. [2] L.J. Alías, M. Dajczer, Constant mean curvature hypersurfaces in warped product spaces, Proc. Edinb. Math. Soc. 50 (2007) 511–526. [3] L.J. Alías, M. Dajczer, Constant mean curvature graphs in a class of warped product spaces, Geom. Dedicata 131 (2008) 173–179. [4] V.I. Arnol’d, Ordinary Differential Equations, translated from Russian by Roger Cooke, Springer-Verlag, Berlin, 1992. [5] R.L. Bishop, B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969) 1–49. [6] A. Caminha, H.F. de Lima, Complete vertical graphs with constant mean curvature in semi-Riemannian warped products, Bull. Belg. Math. Soc. Simon Stevin 16 (2009) 91–105. [7] R. Greene, H. Wu, Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Math., vol. 699, Springer-Verlag, Berlin, 1979.

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