Geometric Inequalities and Balls in CR Sasakian Manifolds with Volume for Sub Riemannian Manifolds

Abstract

The minimal submanifolds with constant mean curvature and of a sphere with bounded second fundamental form are considered. An intrinsic rigidity theorem from minimal submanifolds with parallel mean curvature in a sphere and the log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality were studied. Stochastic completeness, volume growth, connection, curvature and distance comparison theorem for sub-Riemannian manifolds are shown. The sub-Riemannian curvature dimension inequality, volume doubling property, Poincaré inequality and balls in CR Sasakian manifolds are discussed. We classify the closed minimal submanifolds and geometric inequalities for certain submanifolds in pinched Riemannian manifolds.