Integral pinched gradient shrinking $\rho$-Einstein solitons

TL;DR

This paper establishes integral pinching rigidity results for compact gradient shrinking $ ho$-Einstein solitons, expanding understanding of their geometric structure through algebraic curvature estimates and the Yamabe-Sobolev inequality.

Contribution

It provides new integral pinching rigidity theorems for compact gradient shrinking $ ho$-Einstein solitons using advanced curvature estimates and inequalities.

Findings

Proved integral pinching rigidity results for compact gradient shrinking $ ho$-Einstein solitons.

Applied algebraic curvature estimates and Yamabe-Sobolev inequality in the proofs.

Enhanced understanding of the geometric structure of these solitons.

Abstract

The gradient shrinking $\rho$-Einstein soliton is a triple $(M^n,g,f)$ such that $$R_{ij}+f_{ij}=(\rho R+\lambda) g_{ij},$$ where $(M^n,g)$ is a Riemannian manifold, $\lambda>0, \rho\in\mathbb{R}\setminus\{0\}$ and $f$ is the potential function on $M^n$. In this paper, using algebraic curvature estimates and the Yamabe-Sobolev inequality, we prove some integral pinching rigidity results for compact gradient shrinking $\rho$-Einstein solitons.