Rigidity of compact quasi-Einstein manifolds with boundary

Abstract

In this article, we investigate the geometry of compact quasi-Einstein manifolds with boundary. We establish the possible values for the constant scalar curvature of a compact quasi-Einstein manifold with boundary. Moreover, we show that a $3$-dimensional simply connected compact $m$-quasi-Einstein manifold with boundary and constant scalar curvature must be isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{3}_{+}$, or the cylinder $\left[0,\frac{\sqrt{m}}{\sqrtλ}\,π\right]\times\mathbb{S}^2$ with the product metric. For dimension $n=4,$ we prove that a $4$-dimensional simply connected compact $m$-quasi-Einstein manifold $M^4$ with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{4}_{+},$ or the cylinder $\left[0,\frac{\sqrt{m}}{\sqrtλ}\,π\right]\times\mathbb{S}^3$ with the product metric, or the product space $\mathbb{S}^{2}_{+}\times\mathbb{S}^2$ with the doubly warped product metric. Other related results for arbitrary dimensions are also discussed.

Theorems & Definitions (54)

• Remark 1
• Theorem 1
• Remark 2
• Remark 3
• Theorem 2
• Remark 4
• Theorem 3
• Theorem 4
• Corollary 1
• Proposition 1: O'Neil