Integral estimates for the trace of symmetric operators

TL;DR

This paper establishes conditions under which the integral of the trace of a symmetric operator on a manifold immersed in a Hadamard space is either zero or infinite, with applications to minimal submanifolds and curvature bounds.

Contribution

It provides new integral estimates for symmetric operators on immersed manifolds, linking geometric properties to integrability conditions and extending previous results.

Findings

If certain conditions hold, the trace integral is either zero or infinite.

Manifolds with minimal foliation have infinite volume under specified conditions.

Curvature bounds lead to integrability or triviality of functions related to the immersion.

Abstract

Let $\Phi:TM\to TM$ be a positive-semidefinite symmetric operator of class $C^1$ defined on a complete non-compact manifold $M$ isometrically immersed in a Hadamard space $\bar{M}$. In this paper, we given conditions on the operator $\Phi$ and on the second fundamental form to guarantee that either $\Phi\equiv 0$ or the integral $\int_M \mathrm{tr}\,\Phi dM$ is infinite. We will given some applications. The first one says that if $M$ admits an integrable distribution whose integrals are minimal submanifolds in $\bar{M}$ then the volume of $M$ must be infinite. Another application states that if the sectional curvature of $\bar{M}$ satisfies $\bar{K}\leq -c^2$, for some $c\geq 0$, and $\lambda:M^m\to [0,\infty)$ is a nonnegative $C^1$ function such that gradient vector of $\lambda$ and the mean curvature vector $H$ of the immersion satisfy $|H+p\nabla \lambda|\leq (m-1)c \lambda$, for some $p\geq 1$, then either $\lambda\equiv 0$ or the integral $\int_M \lambda^s dM$ is infinite, for all $1\leq s\leq p$.