On Riemannian manifolds with positive weighted Ricci curvature of negative effective dimension

TL;DR

This paper explores Riemannian manifolds with positive weighted Ricci curvature in the context of negative effective dimension, providing examples, eigenvalue analysis, and a splitting theorem under certain conditions.

Contribution

It introduces new results on the structure of manifolds with positive weighted Ricci curvature for negative effective dimension, including eigenvalue minimization and splitting theorems.

Findings

Constructed explicit 1D examples with finite and infinite volume

Identified conditions for the first eigenvalue to attain its minimum

Proved splitting of manifolds as warped products under certain curvature bounds

Abstract

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound $\mathrm{Ric}_{N} \geq K$ with $K>0$ for the negative effective dimension $N<0$. We analyze two $1$-dimensional examples of constant curvature $\mathrm{Ric}_N \equiv K$ with finite and infinite total volumes. We also discuss when the first nonzero eigenvalue of the Laplacian takes its minimum under the same condition $\mathrm{Ric}_N \ge K>0$, as a counterpart to the classical Obata rigidity theorem. Our main theorem shows that, if $N<-1$ and the minimum is attained, then the manifold splits off the real line as a warped product of hyperbolic nature.