Splitting theorem of Gradient $\rho$-Einstein solitons

TL;DR

This paper proves a splitting theorem for gradient -Einstein solitons under certain curvature conditions, advancing understanding of their geometric structure and scalar curvature bounds.

Contribution

It establishes a new splitting theorem for gradient -Einstein solitons with bounded Ricci curvature integral conditions, and derives scalar curvature bounds.

Findings

Gradient -Einstein solitons with bounded Ricci curvature split off a line.

Weighted Laplacian comparison for manifolds with Bakry-Émery curvature.

Boundedness conditions on scalar curvature of gradient -Einstein solitons.

Abstract

In this paper, we have proved a weighted Laplacian comparison of distance function for manifolds with Bakry-\'Emery curvature bounded from below. Next, we have shown that a gradient $\rho$-Einstein soliton with a bounded integral condition on Ricci curvature splits off a line isometrically. Moreover, using this result, we have established some boundedness conditions on scalar curvature of gradient $\rho$-Einstein soliton.