Manfredo do Carmo

Stability of hypersurfaces with constant mean curvature

Let \(x:{\text M}^{n}\rightarrow\,{\text R}^{n+1}\) be an immersion of an orientable, n-dimensional manifold \({\text M}^{n}\) into the euclidean space \({\text R}^{n+1}\). The condition that x has nonzero constant mean curvature H = H 0 is known to be equivalent to the fact that xis a critical point of a variational problem.

Stability of Hypersurfaces of Constant Mean Curvature in Riemannian Manifolds

Hypersurfaces \(M^n\)with constant mean curvature in a Riemannian manifold \(\overline{M}^{n+1}\)display many similarities with minimal hypersurfaces of \(\overline{M}^{n+1}\). They are both solutions to the variational problem of minimizing the area function for certain variations. In the first case, however, the admissible variations are only those that leave a certain volume function fixed (for precise definitions, see Sect. 2). This isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces.

Compact Conformally Flat Hypersurfaces

Roughly speaking, a conformal space is a differentiable manifold \(M^n\)in which the notion of angle of tangent vectors at a point \(p \in M^n\)makes sense and varies differentiably with p; two such spaces are (locally) equivalent if they are rela ted by an angle-preserving (local) diffeomorphism. A conformally flat space is a conformal space locally equivalent to the euclidean space R n. A submanifold of a conformally flat space is said to be conformally flat if so its induced conformal structure: in particular, if the codimension is one, it is called a conformally flat hypersurface. The aim of this paper is to give a description of compact conformally flat hypersurfaces of a conformally flat space. For simplicity, as~ume the ambient space to be R n+1. Then, if \(n \geqslant 4\), a conformally flat hypersurface \({M}^{n} \subset {R}^{n+1}\) 1 can be described as follows. Diffeomorphically, M n is a sphere S n with h1( M) handles attached, where h1 ( M) is the first Betti number of M. Geometrically, it is made up by (perhaps infinitely many) nonumbilic submanifolds of R n+1 that are foliated by complete round (n – 1 )-spheres and are joined through their boundaries to the following three types of umbilic submanifolds of R n+1: (a) an open piece of an n-sphere or an n-plane bounded by round ( n – 1 )-sphere, (b) a round ( n – 1 )-sphere, (c) a point.

Stability of Minimal Surfaces and Eigenvalues of the Laplacian

Let \(x:{\text M}\rightarrow \,\tilde{M}^{n}\) be a minimal immersion of a two-dimensional orientable manifold \({\text M}\) into an n-dimensional Riemannian manifold \(\tilde{M}^{n}.\)

A Hopf Theorem for Ambient Spaces of Dimensions Higher than Three

We consider surfaces M 2 immersed in \( E_{c}^{n} \times \mathbb{R}\), where \( E_{c}^{n}\) is a simply connected n-dimensional complete Riemannian manifold with constant sectional curvature \( E_{c}^{n}\), and assume that the mean curvature vector of the immersion is parallel in the normal bundle. We consider further a Hopf-type complex quadratic form Q on M 2, where the complex structure of M 2 is compatible with the induced metric. It is not hard to check that Q is holomorphic (see [3], p.289). We will use this fact to give a reasonable description of immersed surfaces in \( E_{c}^{n} \times \mathbb{R}\) that have parallel mean curvature vector.

On the first eigenvalue of the linearized operator of ther-th mean curvature of a hypersurface

We generalize Reillys inequality for the first eigenvalue of immersed submanifolds ofIRm+1 and the total (squared) mean curvature, to hypersurfaces ofIRm+1 and the first eigenvalue of the higher order curvatures. We apply this to stability problems. We also consider hypersurfaces in hyperbolic space.

A proof of a general isoperimetric inequality for surfaces

(1.1) Let M be a two-dimensional C2-manifold endowed with a C2-Riemannian metric. We say that M is a generalized surface if the metric in M is allowed to degenerate at isolated points; such points are called singularities of the metric. In this paper we use the method of Fiala-Bol (cf. [12, 9]) to give a proof of the following general isoperimetric inequality.

Complete manifolds with non-negative Ricci curvature and the Caffarelli–Kohn–Nirenberg inequalities

In this paper, we prove that complete open Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to three in which some Caffarelli–Kohn–Nirenberg type inequalities are satisfied are close to the Euclidean space.

Ricci curvature and the topology of open manifolds

In this paper, we prove that an open Riemannian n-manifold with Ricci curvature \({\rm Ric}_M\geq 0\) and \(K_p^{\rm min}\geq K_0> -\infty\) for some \(p\in M\) is diffeomorphic to a Euclidean n-space \(R^n\) if the volume growth of geodesic balls around p is not too far from that of the balls in \(R^n\). We also prove that a complete n-manifold M with \(K_p^{\rm min}\geq 0\) is diffeomorphic to \(R^n\) if \(\lim_{r\rightarrow \infty}\frac{{\rm Vol}[B(p,r)]}{\omega_n r^n}\geq \frac 12\), where \(\omega_n\) is the volume of unit ball in \(R^n\).

Bernstein-type theorems in hypersurfaces with constant mean curvature

By using the nodal domains of some natural function arising in the study of hypersurfaces with constant mean curvature we obtain some Bernstein-type theorems.