Ends of shrinking gradient $\rho$-Einstein solitons

TL;DR

This paper investigates the geometric properties of gradient shrinking $ ho$-Einstein solitons, demonstrating conditions under which their ends are $ ext{varphi}$-non-parabolic and establishing their connectedness at infinity for certain parameter ranges.

Contribution

It proves that all ends of such solitons are $ ext{varphi}$-non-parabolic under specific curvature conditions and shows their connectedness at infinity for a range of $ ho$ values.

Findings

Ends are $ ext{varphi}$-non-parabolic when scalar curvature is bounded and nonnegative.

Solitons are connected at infinity for $ ho ext{ in } [0, 1/2(n-1)]$ with $n ext{ at least } 4$.

Results depend on bounds for scalar curvature and the parameter $ ho$.

Abstract

We prove that all ends of a gradient shrinking $\rho$-Einstein soliton are $\varphi$-non-parabolic, provided $\rho$ is nonnegative and the soliton has bounded and nonnegative scalar curvature, where the weight $\varphi$ is a negative multiple of the potential function. We also show these solitons are connected at infinity for $\rho \in\left[0,1/2(n-1)\right)$, $n\geq4$, and a suitable bound for the scalar curvature.