The structure of stable constant mean curvature hypersufaces

We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature hypersurfaces in space forms. In particular, a complete oriented weakly stable minimal hypersurface in Rn+1, n ≥ 3, must have only one end. Any complete noncompact weakly stable CMC H-hypersurface in the hyperbolic space Hn+1, n = 3, 4, with H2 ≥ 10 9 , 74 , respectively, has only one end. 0 Introduction The classical Bernstein theorem states that a minimal entire graph in R must be planar. This theorem was later generalized to higher dimensions (dimension of the ambient Euclidean space R is no more than 8) by Fleming[Fl], Almgren[A], De Giorgi[Dg], and Simons[S]. In R, n ≥ 8, the examples of nonlinear entire graphs are given by Bombieri, de Giorgi and Giusti [BdGG]. Because of the stability of minimal entire graphs, one is naturally led to the generalization of the classical Bernstein theorem to the question of asking whether all stable minimal hypersurfaces in R are hyperplanes when n ≤ 7. In the case when n = 2, this problem was solved independently by do Carmo and Peng [dCP]; and Fischer-Colbrie and Schoen [FS]. For higher Supported by CNPq of Brazil Supported by CAPES and CNPq of Brazil. 1 dimension, this problem is still open. On the other hand, there are some results about the structure of stable minimal hypersurfaces in all R. For instance, H. Cao, Y. Shen and S. Zhu [CSZ] proved that a complete stable minimal hypersurface in R, n ≥ 3, must have only one end. If the ambient manifold is not the Euclidean space, Fischer-Colbrie and Schoen [FS] gave a classification for complete oriented stable minimal surfaces in a complete oriented 3-manifold of nonnegative scalar curvature. Recently, Li and Wang [LW1] showed that a complete noncompact properly immersed stable minimal hypersurface in a complete manifold of nonnegative sectional curvature must either have only one end or be totally geodesic and a product of a compact manifold with nonnegative sectional curvature and R. In this paper we study hypersurfaces with constant mean curvature H . Let us now fix terminologies and notations so as to our theorems. In the sequel we will abbreviate constant mean curvature hypersurfces by calling them CMC H−hypersurfaces and will allow H to vanish (hence the need of putting H here). Instead of the usual stability, we will consider a weaken form of stability, which is in fact the natural one for CMC H−hypersurfaces in case H 6= 0. Intuitively, a CMC hypersurface is weakly stable if the second variations are nonnegative for all compactly supported enclosed-volumepreserving variations (see Definition 1.1 and Remark 1.1). This concept of weakly stable CMC hypersurfaces was introduced by Barbosa, do Carmo and Eschenburg [BdCE], to accounts for the fact that spheres are stable (see [BdCE]). This weak stability comes naturally from the phenomenon of soap bubbles and is related to isoperimetric problems. In [dS], da Silveira studied complete noncompact weakly stable CMC surfaces in R or the hyperbolic space H. He proved that complete weakly stable CMC surfaces in R 3 are planes and hence generalized the corresponding result of do Carmo and Peng [dCP], Fischer-Colbrie and Schoen [FS]. For H he shows that only horospheres can occur when constant mean curvature |H| ≥ 1. For higher dimensions, very little is known about complete noncompact weakly stable CMC hypersurfaces. In this paper, we study the global behavior of weakly stable CMC hypersurface (including minimal case). First, we obtain Theorem 0.1. (Th.3.4) Let N, n ≥ 5, be a complete Riemannian manifold and M be a complete noncompact weakly stable immersed CMC Hhypersurface in N . If one of the following cases occurs, (1) when n = 5, the sectional curvature of N is nonnegative and H 6= 0; 2 (2) when n ≥ 6, the sectional curvature K̃ of N satisfies K̃ ≥ τ > 0 and H ≤ 4(2n−1) n2(n−5)τ, for some number τ > o; (3) when n ≥ 6, the sectional curvature and the Ricci curvature of N satisfy K̃ ≥ 0, R̃ic ≥ τ > 0, for some number τ > 0, and H = 0, then M has only one end. The reason for the restriction on dimensions of CMC hypersurfaces in the above theorem is that there are some nonexistence results (see the proof of this theorem for detail). Theorem 0.1 has the following examples: complete noncompact weakly stable CMC H-hypersurfaces in the standard sphere S with H 6= 0; or in the standard sphere S, n ≥ 6 with H ≤ 4(2n−1) n2(n−5) . Actually, Theorem 0.1 is a special case of the more general Theorem 3.3, which also implies that Theorem 0.2. (Cor.3.3) Any complete noncompact weakly stable CMC Hhypersurface in the hyperbolic space H, n = 3, 4, with H ≥ 10 9 , 7 4 , respectively, has only one end. Next we consider complete weakly stable minimal hypersurfaces in R, n ≥ 3, and generalize the results of Cao, Shen and Zhu as follows: Theorem 0.3. (Th.3.2) A complete oriented weakly stable minimal hypersurface in R, n ≥ 3, must have only one end. With this theorem, we obtain Corollary 0.1. (Cor.3.1) A complete oriented weakly stable immersed minimal hypersurface in R, n ≥ 3, with finite total curvature (i.e., ∫ M |A|n < ∞) is a hyperplane. Finally, we study the structure of weakly stable CMC hypersurfaces according to the parabolicity or nonparabolicity of M . We obtain the following results: Theorem 0.4. (Th.5.2) Let N be a complete manifold of bounded geometry and M be a complete noncompact weakly stable CMC H-hypersurface immersed in N . If the sectional curvature of N is bounded from below by −H2 and M is parabolic, then it is totally umbilic and has nonnegative sectional curvature. Furthermore, either (1) M has only one end; or (2) M = R × P with the product metric, where P is a compact manifold of nonnegative sectional curvature. 3 Theorem 0.5. (Th.5.3) Let N be a complete Riemannian manifold and M be a complete noncompact weakly stable CMC H-hypersurface immersed in N . If M is nonparabolic, and R̃ic(ν) + R̃ic(X)− K̃(X, ν) ≥ n 2(n− 5) 4 H, ∀X ∈ TpM, |X| = 1, p ∈ M, then it has only one nonparabolic end, where K̃ and R̃ic denote the sectional and Ricci curvatures of N , respectively; ν denotes the unit normal vector field of M . In some of recent works, the structure of stable (i.e., strongly stable) minimal hypersurfaces was studied by means of harmonic function theory (see [CSZ], [LW], [LW1]). The same approach can be used in the study of weakly stable CMC hypersurfaces. However, a significant difference between weakly stable and strongly stable cases lies in the choice of test functions. When one deals with weak stability, the test functions f must satisfy ∫ M f = 0. In this paper, we successfully construct the required test functions by using the properties of harmonic functions (Theorem 3.1 and Proposition 4.1). Combining our construction and the approach in [LW], [LW1], we are able to discuss the global behavior of weakly stable CMC hypersurfaces. In Theorem 3.1, we obtain the nonexistence of nonconstant bounded harmonic functions with finite Dirichlet integral on weakly stable CMC hypersurfaces. This theorem enable us to study the uniqueness of ends. In Proposition 4.1, we discuss a property of Schrödinger operator on parabolic manifolds which can be applied to study weakly stable CMC hypersurfaces with parabolicity. Besides, different from minimal hypersurfaces, CMC hypersurfaces with H 6= 0 have the curvature estimate depending on H , which causes dimension restriction in the results. The rest of this paper is organized as follows: in Section 1 we give some definitions and facts as preliminaries; in Section 2, we first discuss volume growth of the ends of complete noncompact hypersurfaces with mean curvature vector field bounded in norm, and then study nonparabolicity of the ends of CMC hypersurfaces with stability; in Section 3, we use harmonic functions to study the uniqueness of ends of complete noncompact weakly stable CMC hypersurfaces; in Section 4, we give a property of Schrödinger operator on parabolic manifolds; in the last section (Section 5), we discuss the structure of complete noncompact weakly stable CMC hypersurfaces. The results on minimal case in this paper has been announced in [CCZ]. 4 Acknowledgements. One part of this work was done while the third author was visiting the Department of Mathematics, University of California, Irvine. He wishes to thank the department for hospitality. The authors would like to thank Peter Li for some conversations. 1 Preliminaries We recall some definitions and facts in this section. Let N be an oriented (n + 1)-dimensional Riemannian manifold and let i : M → N be an isometric immersion of a connected n-dimensional manifold M with constant mean curvature H . We assume M is orientable. When H is nonzero, the orientation is automatic. Throughout this paper, K̃, R̃ic, K, and Ric denote the sectional, Ricci curvatures of N , the sectional, Ricci curvature of M respectively. ν denotes the unit normal vector field of M . |A| is the norm of the second fundamental form A. Bp(R) will denote the intrinsic geodesic ball in M of radius R centered at p. We have Definition 1.1. There are two cases. In the case H 6= 0, the immersion i is called stable or weakly stable if