Deforming Convex Hypersurfaces to a Hypersurface with Prescribed Harmonic Mean Curvature

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Linear Weingarten hypersurfaces in a unit sphere

In this paper, by modifying Cheng-Yau$'$s technique to complete hypersurfaces in $S^{n+1}(1)$, we prove a rigidity theorem under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related which improve the result of [H. Li, Hypersurfaces with constant scalar curvature in space forms, {em Math. Ann.} {305} (1996), 665--672].  

No Mass Drop for Mean Curvature Flow of Mean Convex Hypersurfaces

Abstract. A possible evolution of a compact hypersurface in R by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a consequence that no mass drop can occur along such a flow. As a further ap...

Convex hypersurfaces of prescribed curvatures

For a smooth strictly convex closed hypersurface Σ in R, the Gauss map n : Σ → S is a diffeomorphism. A fundamental question in classical differential geometry concerns how much one can recover through the inverse Gauss map when some information is prescribed on S ([27]). This question has attracted much attention for more than a hundred years. The most notable example is probably the Minkowski...

Hypersurfaces of Prescribed Scalar Curvature in Lorentzian Manifolds

The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers. 0. Introduction Consider the problem of finding a closed hypersurface of prescribed curvature F in a globally hyperbolic (n+1)-dimensional Lorentzian manifold N having a compact Cauchy hypersurface S0. To be more precise, let Ω be a connected op...