A generalization of the Cai--Galloway splitting theorem to smooth metric measure spaces

TL;DR

This paper extends the Cai-Galloway splitting theorem to smooth metric measure spaces, providing sharp conditions and new insights into the geometry of such spaces, with applications to conformally compact quasi-Einstein metrics.

Contribution

It generalizes the splitting theorem to a broader setting and sharpens existing curvature conditions, offering new geometric and spectral insights.

Findings

The mean curvature assumption is sharp.

The Riemannian curvature condition cannot be relaxed.

Necessary conditions for boundary connectedness and spectral maximality.

Abstract

We generalize the splitting theorem of Cai-Galloway for complete Riemannian manifolds with $\Ric\geq-(n-1)$ admitting a family of compact hypersurfaces tending to infinity with mean curvatures tending to $n-1$ sufficiently fast to the setting of smooth metric measure spaces. This result complements and provides a new perspective on the splitting theorems recently proven by Munteanu-Wang and Su-Zhang. We show that the mean curvature assumption in our result is sharp, which also provides an example showing that the assumption $R\geq-(n-1)$ in the Munteanu-Wang splitting theorem for expanding gradient Ricci solitons cannot be relaxed to $R>-n$. We also use our result to study a certain class of conformally compact quasi-Einstein metrics, giving, as generalizations of respective results of Cai-Galloway and Lee, necessary conditions for the boundary to be connected and for the bottom of the spectrum of the weighted Laplacian to be maximal.