Rigidity Theorems of conformal class on compact manifolds with boundary

TL;DR

This paper establishes rigidity theorems for metrics within a conformal class on compact manifolds with boundary, under integral curvature conditions, without requiring eigenvalue constraints.

Contribution

It introduces new rigidity results for conformal metrics on manifolds with boundary, removing the need for eigenvalue conditions.

Findings

Rigidity theorems for conformal metrics with fixed boundary

Integral conditions on scalar and mean curvatures suffice

No eigenvalue conditions are necessary

Abstract

Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean curvatures on the boundary. No condition on the first eigenvalues of operators is need.