Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons

TL;DR

This paper investigates the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons, establishing their connectedness at infinity and conditions for compactness under curvature assumptions.

Contribution

It proves that such solitons are connected at infinity and, under positive curvature conditions, must be compact, extending known results from Kähler and Ricci solitons.

Findings

Complete gradient shrinking Sasaki-Ricci solitons are connected at infinity.

Positive curvature conditions imply compactness of these solitons.

Results generalize key theorems from Perelman, Naber, and Munteanu-Wang.

Abstract

In this paper, we study the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons. We first prove that they must be connected at infinity. This is a Sasaki analogue of gradient shrinking K\"ahler-Ricci solitons. Secondly, with the positive sectional curvature or positive transverse holomorphic bisectional curvature, we show that they must be compact. All results are served as a generalization of Perelman in dimension three, of Naber in dimension four, and of Munteanu-Wang in all dimensions, respectively.