Locally constrained flows and geometric inequalities in sphere

Abstract

In this paper, we uncover an intriguing algebra property of an element symmetric polynomial. By this property, we establish the longtime existence and convergence of a locally constrained flow, thereby some families of geometric inequalities in sphere can be derived. Meanwhile, a new family of ``three terms'' geometric inequalities involving two weighted curvature integrals and one quermassintegral are proved. Unlike hyperbolic spaces, we also obtain an inverse weighted geometric inequality in sphere.

There are some errors in the case of hyperbolic space in v1. We removed the corresponding part, and added a locally constrained flow in sphere