Rigidity of five-dimensional quasi-Einstein manifolds with constant scalar curvature

TL;DR

This paper proves the rigidity of five-dimensional non-trivial compact quasi-Einstein manifolds with boundary and constant scalar curvature in the case where k=3, extending previous classifications for other values of k.

Contribution

It establishes the rigidity of the case k=3 for five-dimensional quasi-Einstein manifolds with constant scalar curvature, completing the classification for all relevant k values.

Findings

The case k=3 is rigid for the specified manifolds.

Previous cases k=0 and k=4 are fully classified.

Provides a complete understanding of scalar curvature conditions in this context.

Abstract

Let $(M^5,g)$ be a five-dimensional non-trivial simply-connected compact quasi-Einstein manifold with boundary. If $M$ has constant scalar $R$, Johnatan Costa, Ernani Ribeiro Jr, and Detang Zhou show that $R$ = $((m-5)k+20)/(m-k+4)\lambda$ for some $k\in\{0,2,3,4\}$. Both cases of $k=0$ and $k=4$ are already classified. In this paper we will prove that the case $k=3$ is rigid.