Warped product rigidity

TL;DR

This paper investigates the solution space of a Hessian-based linear system on manifolds, revealing that a high-dimensional solution space implies a rigid warped product structure, with applications to Einstein metrics.

Contribution

It establishes a rigidity result linking the solution space dimension to warped product structures and provides a uniqueness theorem for Ricci curvature prescription on such manifolds.

Findings

High-dimensional solution space implies warped product rigidity

Uniqueness of Ricci curvature prescription on warped products

Application to Einstein structures

Abstract

In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. We obtain a uniqueness result for prescribing the Ricci curvature of a warped product manifold over a fixed base. As an application, this warped product structure will be used to study warped product Einstein structures in "Uniqueness of warped product Einstein metrics and applications".