Weimin Sheng

Interior curvature bounds for a class of curvature equations

We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Monge-Amp` ere type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C 1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.

Prescribing the symmetric function of the eigenvalues of the Schouten tensor

In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is nonnegative-definite.

GEOMETRY OF COMPLETE HYPERSURFACES EVOLVED BY MEAN CURVATURE FLOW

Some geometric behaviours of complete solutions to mean curvature flow before the singularities occur are studied. The author obtains the estimates of the rate of the distance between two fixed points and the derivatives of the second fundamental form. By use of a new maximum principle, some geometric properties at infinity are obtained.

On the affine diameter of closed convex hypersurfaces

In this paper we prove that the affine diameter of any closed uniformly convex hypersurface in Euclidean space enclosing finite volume is bounded from above.

Variational properties of quadratic curvature functionals

In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold M of dimension n ≥ 3 on the space of Riemannian metrics on M consisting of metrics with unit volume. We study the stability of these functionals at the metric with constant sectional curvature as its critical point.