Conformal invariants for the zero mode equation

TL;DR

This paper establishes a sharp inequality relating the zero mode equation on spin manifolds to the Yamabe constant, characterizes the equality case, and explores geometric implications including Sasaki-Einstein structures.

Contribution

It provides a simple proof of a sharp inequality involving the zero mode equation and classifies the equality case with geometric and spinorial conditions.

Findings

The inequality \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]) is sharp.

Equality holds iff the solution is a Killing spinor and the manifold is Sasaki-Einstein.

Characterization of solutions in terms of Reeb fields and vacuum states.

Abstract

For non-trivial solutions to the zero mode equation on a closed spin manifold \[D \varphi=iA\cdot \varphi,\] we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$ is the Yamabe constant of $(M,g)$, which was obtained by Frank-Loss and Reuss. Then we classify completely the equality case by proving that equality holds if and only if $\varphi$ is a Killing spinor, and if and only if $(M,g)$ is a Sasaki-Einstein manifold with $A$ (up to scaling) as its Reeb field and $\varphi$ a vacuum up to a conformal transformation. More generalizations have been also studied.