Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Abstract

Using Rauch's comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li-Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch's comparison theorem, the proofs mainly rely on the first variation formula, thus are valid for general varifolds.

Poincaré inequality on minimal graphs over manifolds and applications

“…The Sobolev inequality on minimal submanifolds can be generalized from Euclidean space to Riemannian manifolds. In [41], Hoffman-Spruck obtained the Sobolev inequality on a minimal submanifold M in a manifold N , with some geometric restrictions involving the volume of M , the sectional curvatures of N and the injectivity radius of N (see [43] [56] for more results). Recently, Brendle [8] proved the Sobolev inequality on minimal submanifolds in manifolds of nonnegative sectional curvature and Euclidean volume growth.…”