Sharp metric obstructions for quasi-Einstein metrics

TL;DR

This paper develops sharp tensorial obstructions for the existence of quasi-Einstein metrics on manifolds using tractor calculus, extending previous results to higher dimensions and related geometric structures.

Contribution

It introduces an analogue of the Weyl tractor for smooth metric measure spaces and derives new polynomial tensor invariants that obstruct quasi-Einstein metrics and related potentials.

Findings

Derived tensorial obstructions for quasi-Einstein metrics.

Extended obstructions to static potentials and gradient Ricci solitons.

Generalized previous 2D results to higher dimensions.

Abstract

Using the tractor calculus to study smooth metric measure spaces, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the Weyl tractor $W$ to the setting of smooth metric measure spaces. The obstructions we obtain can be realized as tensorial invariants which are polynomial in the Riemann curvature tensor and its divergence. By taking suitable limits of their tensorial forms, we then find obstructions to the existence of static potentials, generalizing to higher dimensions a result of Bartnik and Tod, and to the existence of potentials for gradient Ricci solitons.