Mean curvature rigidity of horospheres, hyperspheres, and hyperplanes
- PDF / 291,193 Bytes
- 6 Pages / 439.37 x 666.142 pts Page_size
- 67 Downloads / 233 Views
Archiv der Mathematik
Mean curvature rigidity of horospheres, hyperspheres, and hyperplanes Rabah Souam
Abstract. We prove that horospheres, hyperspheres, and hyperplanes in a hyperbolic space Hn , n ≥ 3, admit no perturbations with compact support which increase their mean curvature. This is an extension of the analogous result in the Euclidean spaces, due to M. Gromov, which states that a hyperplane in a Euclidean space Rn admits no mean convex perturbations with compact supports. Mathematics Subject Classification. 53C21, 53C24, 53C40. Keywords. Mean curvature, Mean convexity, Tangency principle.
The content of the present note is motivated by the following nice recent result of Gromov [1]. Theorem 1. A hyperplane in a Euclidean space Rn cannot be perturbed on a compact set so that its mean curvature satisfies H ≥ 0. This statement is reminiscent of the scalar curvature rigidity of the flat metric, a corollary of the positive mass theorem, which asserts that the flat metric on Rn cannot be modified on a compact set so that its scalar curvature verifies S ≥ 0. Actually, Gromov [1] derived Theorem 1 from the nonexistence of Zn -invariant metrics with positive scalar curvature on Rn . He also gave another direct argument using a symmetrization process. In what follows, we give another proof of Theorem 1 and an extension to the hyperbolic spaces using a simple argument. More precisely, we prove the following: Theorem 2. Let M denote a horosphere, a hypersphere, or a hyperplane in a hyperbolic space Hn , n ≥ 3, and HM ≥ 0 its (constant) mean curvature. Let Σ be a connected properly embedded C 2 -hypersurface in Hn which coincides with M outside a compact subset of Hn . If the mean curvature of Σ is ≥ HM , then Σ = M.
R. Souam
Arch. Math.
We recall that a hyperplane in Hn is a complete totally geodesic hypersurface and a hypersphere is a connected component of the set of points at a fixed distance from a hyperplane. The proof uses the tangency principle which goes back to E. Hopf. We recall it with some details in what follows. We first fix some notations and conventions. Let Σ be an embedded C 2 -hypersurface, with a global unit normal ν, in a smooth complete n-dimensional Riemannian manifold M . We denote by σ the second fundamental form of Σ, which is defined as follows σp (u, v) = − < ∇u ν, v >
for p ∈ Σ, u, v ∈ Tp Σ.
Here ∇ denotes the Levi–Civita connection on M . The shape operator S of Σ is defined as follows Sp u = −∇u ν
for p ∈ Σ, u ∈ Tp Σ,
and the (normalized) mean curvature of Σ is the function 1 trace S. H= n−1 Note that these definitions depend on the choice of the unit normal field ν. The mean curvature vector field H := Hν is, instead, independent of the choice of the unit normal field. With our conventions, the mean curvature of a unit sphere in the Euclidean space with respect to its interior unit normal is equal to 1. Let p ∈ Σ and consider local coordinates (x1 , . . . , xn ) around p in M so that Tp Σ = Rn−1 × {0} and ∂x∂n (0) = ν(p). An open neighborhood U of p in Σ is the graph, in these coordinat
Data Loading...
