Cohomogeneity-One Quasi-Einstein Metrics

TL;DR

This paper investigates $G$-invariant quasi-Einstein metrics on cohomogeneity one manifolds derived from homogeneous spaces, providing estimates on their blow-up behavior and constructing metrics satisfying prescribed boundary conditions.

Contribution

It introduces new estimates on blow-up rates and demonstrates the existence of quasi-Einstein metrics meeting arbitrary boundary conditions under symmetry assumptions.

Findings

Derived bounds on blow-up rates near singularities.

Established existence of metrics with arbitrary boundary conditions.

Analyzed behavior of quasi-Einstein metrics under symmetry constraints.

Abstract

Let $G/H$ be a connected, simply connected homogeneous space of a compact Lie group $G$. We study $G$-invariant quasi-Einstein metrics on the cohomogeneity one manifold $G/H\times (0,1)$ imposing the so-called monotypic condition on $G/H$. We obtain estimates on the rate of blow-up for these metrics near a singularity under a mild assumption on $G/H$. Next, we demonstrate that we can find quasi-Einstein metrics satisfying arbitrary $G$-invariant Dirichlet conditions.