Yamabe metrics, Fine solutions to the Yamabe flow, and local L1-stability

TL;DR

This paper investigates the existence and stability of Yamabe metrics with zero scalar curvature on complete Riemannian manifolds, demonstrating global solutions to the Yamabe flow without curvature restrictions and analyzing local L1-stability.

Contribution

It establishes the existence of complete Yamabe metrics with zero scalar curvature without initial curvature assumptions and proves local L1-stability of the Yamabe flow on manifolds with non-negative Ricci curvature.

Findings

Existence of global fine solutions to the Yamabe flow under minimal initial conditions.

L1-stability of the Yamabe flow on manifolds with non-negative Ricci curvature.

Discussion of Yamabe metrics with zero scalar curvature on model spaces.

Abstract

In this paper, we study the existence of complete Yamabe metric with zero scalar curvature on an n-dimensional complete Riemannian manifold $(M,g_0)$, $n\geq 3$. Under suitable conditions about the initial metric, we show that there is a global fine solution to the Yamabe flow. The interesting point here is that we have no curvature assumption about the initial metric. We show that on an n-dimensional complete Riemannian manifold $(M,g_0)$ with non-negative Ricci curvature, $n\geq 3$, the Yamabe flow enjoys the local $L^1$-stability property from the view-point of the porous media equation. Complete Yamabe metrics with zero scalar curvature on an n-dimensional Riemannian model space are also discussed.