Monotonicity formulae, vanishing theorems and some geometric applications

TL;DR

This paper develops monotonicity formulae using stress energy tensors for vector bundle-valued p-forms on Riemannian and Kähler manifolds, leading to vanishing theorems, geometric applications like Ricci flatness, and classical volume monotonicity results.

Contribution

It introduces new monotonicity formulae for p-forms satisfying conservation laws on manifolds with exhaustion functions, extending classical results and deriving novel geometric consequences.

Findings

Monotonicity formulae for Ricci form and volume of minimal submanifolds.

Vanishing theorems under growth conditions on energy.

Bernstein type results for minimal submanifolds with weaker assumptions.

Abstract

Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. K\"ahler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae under suitable growth conditions on the energy of the p-forms. As an application, we establish a monotonicity formula for the Ricci form of a K\"ahler manifold of constant scalar curvature and then get a growth condition to derive the Ricci flatness of the K\"ahler manifold. In particular, when the curvature does not change sign, the K\"ahler manifold is isometrically biholomorphic to C^m. Another application is to deduce the monotonicity formulae for volumes of minimal submanifolds in some outer spaces with suitable exhaustion functions. In this way, we recapture the classical volume monotonicity formulae of minimal submanifolds in Euclidean spaces. We also apply the vanishing theorems to Bernstein type problem of submanifolds in Euclidean spaces with parallel mean curvature. In particular, we may obtain Bernstein type results for minimal submanifolds, especially for minimal real K\"ahler submanifolds under weaker conditions.