Rigidity of Homogeneous Gradient Soliton Metrics and Related Equations

TL;DR

This paper establishes new rigidity results for homogeneous spaces supporting solutions to equations involving the Hessian and invariant tensors, extending previous findings on Ricci solitons and Einstein metrics.

Contribution

It generalizes earlier rigidity results to broader classes of equations and applies to homogeneous gradient solitons and conformally Einstein metrics.

Findings

Rigidity results for solutions to Hessian-involving equations on homogeneous spaces

Extension of previous results to trace-free systems

New structure theorem for homogeneous conformally Einstein metrics

Abstract

We prove structure results for homogeneous spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function and an invariant 2-tensor. We also consider trace-free versions of these systems. Our results generalize earlier rigidity results for gradient Ricci solitons and warped product Einstein metrics. In particular, our results apply to homogeneous gradient solitons of any invariant curvature flow and give a new structure result for homogeneous conformally Einstein metrics.