Complete gradient shrinking Ricci solitons have finite topological type

TL;DR

This paper proves that under certain curvature and geometric bounds, complete gradient shrinking Ricci solitons have finite topological type, extending understanding of their topological structure in Riemannian geometry.

Contribution

It establishes conditions under which complete gradient shrinking Ricci solitons possess finite topological type, linking curvature bounds to topological finiteness.

Findings

Finite topological type under positive Bakry-Émery Ricci tensor lower bound.

Finite topological type for shrinking Ricci solitons with bounded scalar curvature.

Topological finiteness results depend on curvature and injectivity radius bounds.

Abstract

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-\'{E}mery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero. Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded.