Total Mean Curvature, Scalar Curvature, and a Variational Analog of Brown–York Mass

Lipschitzian Mappings and Total Mean Curvature

For a smooth closed surface C in E3 the classical total mean curvature is defined by M(C) = ¿/(«i + k2) do(p), where kx, k2 are the principal curvatures at p on C. If C is a polyhedral surface, there is a well known discrete version given by M(C) = IE/,(w a,), where 1¡ represents edge length and a, the corresponding dihedral angle along the edge. In this article formulas involving differentials...

Total Mean Curvature and Closed Geodesics

The proofs and applications are based on a Riemannian version of Gromov’s non-squeezing theorem and classical integral geometry. Given a convex surface Σ ⊂ R and a point q in the unit sphere S we denote by UΣ(q) the perimeter of the orthogonal projection of Σ onto a plane perpendicular to q. We obtain a function UΣ on the sphere which is clearly continuous, even, and positive. Let us denote the...

Mean Curvature Blowup in Mean Curvature Flow

In this note we establish that finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M t →֒ (N, h) are characterised by the blow up of the mean curvature.

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

O ct 2 00 3 Prescribed Scalar Curvature with Minimal Boundary Mean Curvature on S 4

This paper is devoted to the prescribed scalar curvature under minimal boundary mean curvature on the standard four dimensional half sphere. Using topological methods from the theory of critical points at infinity, we prove some existence results.