Mean Curvature In Spaces Research Articles

Constant positive 2-mean curvature hypersurfaces

Hypersurfaces of constant $2$-mean curvature in spaces of constant sectional curvature are known to be solutions to a variational problem. We extend this characterization to ambient spaces which are Einstein. We then estimate the $2$-mean curvature of certain hypersurfaces in Einstein manifolds. A consequence of our estimates is a generalization of a result, first proved by Chern, showing that there are no complete graphs in the Euclidean space with positive constant $2$-mean curvature.

Complete Hypersurfaces with Constant Mean Curvature and Non-Negative Sectional Curvatures

We classify the complete and non-negatively curved hypersurfaces of constant mean curvature in spaces of constant sectional curvature.

Complete hypersurfaces with constant mean curvature and nonnegative sectional curvatures

We classify the complete and non-negatively curved hypersurfaces of constant mean curvature in spaces of constant sectional curvature.

ESTIMATES FOR SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS CONNECTED WITH PROBLEMS OF GEOMETRY "IN THE LARGE"

The following questions are presented in this paper: A geometric method for obtaining two-sided estimates for general quasilinear elliptic equations and its applications to problems of the calculus of variations and the problem of recovering a hypersurface from its mean curvature in spaces of constant curvature. Estimates of the modulus of the gradient for a hypersurface with boundary in a Riemannian space by means of its mean curvature and the metric tensor of the space. Estimates of the modulus of the gradient of a hypersurface depending on the distance of a point from the boundary and its mean curvature in Euclidean space. Estimates of these three types are of independent interest and play a fundamental role in the proofs of existence theorems for a hypersurface with prescribed mean curvature in Riemannian spaces. Bibliography: 3 items.