The Existence of Convex Body with Prescribed Curvature Measures

The Existence of Convex Body with Prescribed Curvature Measures

Existence of Convex Hypersurfaces with Prescribed Gauss-kronecker Curvature

Let f(x) be a given positive function in Rn+1. In this paper we consider the existence of convex, closed hypersurfaces X so that its GaussKronecker curvature at x ∈ X is equal to f(x). This problem has variational structure and the existence of stable solutions has been discussed by Tso (J. Diff. Geom. 34 (1991), 389–410). Using the Mountain Pass Lemma and the Gauss curvature flow we prove the ...

On the Existence and Regularity of Hypersurfaces of Prescribed Gauss Curvature with Boundary

In this paper we study the Dirichlet problem for some Monge-Ampère type equations on S, which naturally arise in some geometric problems. The result then is applied to prove the existence of hypersurfaces in R of prescribed Gauss-Kronecker curvature and with fixed boundary.

Existence of Metrics with Prescribed Scalar Curvature on the Volume Element Preserving Deformation

In this paper,we obtain two results on closed Reimainnian manifold M × [0, T ].When T is small enough,to any prescribed scalar curvature, the existence and uniqueness of metrics are obtained on the volume element preserving deformation.When T is large and the given scalar curvature is small enough,the same result holds.

The Christoffel-minkowski Problem Iii: Existence Problem for Curvature Measures

Curvature measure is one of the basic notion in the theory of convex bodies. Together with surface area measures, they play fundamental roles in the study of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. Let Ω is a bounded convex body in R with C2 boundary M , the corresponding curvature measures and surface area measures of ...