Parabolic Dimensional Reductions of 11D Supergravity

TL;DR

This paper introduces a novel approach to 11D supergravity solutions by formulating a parabolic flow involving Ricci flow coupled with scalar and form fields, enabling PDE techniques to find solutions.

Contribution

It constructs specific ansatze linking 11D supergravity solutions to a coupled Ricci flow, providing new methods for analyzing these solutions using PDE techniques.

Findings

Established Shi-type estimates for the flow

Derived criteria for long-time existence of the flow

Connected supergravity solutions to parabolic PDE analysis

Abstract

Ansatze are constructed under which the solutions of 11D supergravity must be stationary points of a parabolic flow on a Riemannian manifold $M^{10-p}$. This parabolic flow turns out to be the Ricci flow coupled to a scalar field, a $(3-p)$-form, and a $4$-form. This allows the introduction of techniques from parabolic partial differential equations to the search of solutions to 11D supergravity. As a first step, Shi-type estimates and criteria for the long-time existence of the flow are established.