Growth of the eigensolutions of Laplacians on Riemannian manifolds II: positivity of the initial energy

TL;DR

This paper introduces a new method using energy functions to analyze eigen-solutions of Laplacians on Riemannian manifolds, establishing positivity of initial energy and deriving growth properties of solutions.

Contribution

It presents a novel approach to prove initial energy positivity, simplifying the analysis of eigen-solution growth on Riemannian manifolds.

Findings

Positivity of initial energy for eigen-solutions established

Simplified method for analyzing eigen-solution growth

Enhanced understanding of Laplacian eigenfunctions on manifolds

Abstract

In this paper, energy function is used to investigate the eigen-solutions of $-\Delta u+ Vu=\lambda u$ on the Riemannian manifolds. We give a new way to prove the positivity of the initial energy of energy function, which leads to a simple way to obtain the growth of eigen-solutions.