Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold with Boundary

TL;DR

This paper investigates how nontrivial conformal vector fields influence the geometry of compact Riemannian manifolds with boundary, providing characterizations of hemispheres and proving the cosmic no-hair conjecture under certain conditions.

Contribution

It offers new characterizations of hemispheres with constant curvature using the de-Rham Laplacian and Fischer-Marsden equation, and proves the cosmic no-hair conjecture under an integral condition.

Findings

Characterization of hemispheres via Fischer-Marsden equation

Proof of the cosmic no-hair conjecture under specific integral conditions

Analysis of conformal vector fields on manifolds with boundary

Abstract

Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with boundary. In this article, we study the effects of the presence of a nontrivial conformal vector field on $(M^n,g)$. We used the wekk-known de-Rham Laplace operator and a nontrivial solution of the famous Fischer-Marsden differential equation to provide two characterizations of the hemisphere $\mathbb{S}^{n}_{+}(c)$ of constant curvature $c>0.$ As a consequence of the characterization using the Fischer-Marsden equation, we prove the cosmic no-hair conjecture under a given integral condition.