A Weighted Llarull Type Theorem and its applications

TL;DR

This paper extends Llarull's scalar curvature rigidity theorem to weighted manifolds with P-scalar curvature, providing new geometric inequalities and applications to product manifolds.

Contribution

It introduces a weighted Llarull type theorem for P-scalar curvature, generalizing classical results and linking to Listing's work.

Findings

Proves a refined Llarull theorem for P-scalar curvature.

Establishes a Llarull type theorem for manifolds of the form S^k x T^{n-k}.

Provides new tools for scalar curvature rigidity in weighted manifolds.

Abstract

This paper generalizes Llarull's classical scalar curvature rigidity theorem to the setting of weighted manifolds with P-scalar curvature. More precisely, we prove the refinement of Llarull's theorem for P-scalar curvature, which is similar to Listing's work \cite{listing2010scalar}. As an application, we establish a Llarull type theorem in the form of $\mathbb{S}^k\times\mathbb{T}^{n-k}$.