A Spinorial Perelman's Functional: Critical Points and Gradient Flow

TL;DR

This paper introduces a new energy functional on spin manifolds that unifies several known functionals, analyzes its critical points, and studies a gradient flow with proven short-time existence and uniqueness.

Contribution

It defines a novel spinorial energy functional unifying existing functionals, characterizes its critical points, and develops a gradient flow with well-posedness results.

Findings

Critical points are twisted Ricci solitons and eigen-spinsors.

Derived the first variation formula of the functional.

Established short-time existence and uniqueness of the gradient flow.

Abstract

In this article, we introduce an energy functional on closed Riemannian spin manifolds which unifies Perelman's W- and F-functionals, Baldauf-Ouzch's E-functional, and Dirchlet energy for spinors. We compute its first variation formula, and show that its critical points under natural constraints are twisted Ricci solitons and eigen-spinsors of the weighted Dirac operator. We introduce a negative L^2-gradient flow of this functional, and establish its short-time existence and uniqueness via contraction mapping methods.