A Weighted Llarull Type Theorem and its applications

TL;DR

The paper extends Llarull's scalar curvature rigidity to weighted manifolds by introducing $R_f=R+2\Delta f-|\nabla f|^2$ and a weighted spinorial framework. It proves a weighted Llarull-type rigidity: if a smooth map $h:(M,g)\to(\mathbb{S}^n,g_{\mathbb{S}^n})$ has nonzero degree and $R_f\ge n(n-1)\|\wedge^2 h_\ast\|$, then after a positive scale the metric isometrically maps to the sphere and the density $f$ is constant; equality cases are analyzed via weighted Dirac operators. The results are extended to warped $\mathbb{T}^k$-extensions and to Llarull-type rigidity for products $\mathbb{S}^k\times\mathbb{T}^{n-k}$, using a combination of index theory, foliation techniques, and dimension-reduction arguments. Collectively, the work broadens rigidity phenomena to the setting of weighted geometry, with implications for geometric analysis on manifolds with density.

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 $\mathbb{S}^k\times\mathbb{T}^{n-k}$.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Corollary 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10