Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows

TL;DR

This paper introduces a synthetic scalar curvature concept for Riemannian manifolds and metric spaces, linking it to Gaussian integrals, and characterizes Ricci flows as super Ricci flows with minimal integral curvature.

Contribution

It defines a new synthetic scalar curvature via Gaussian integrals and connects Ricci flows to minimal super Ricci flows through this curvature functional.

Findings

Explicit calculation of integral scalar curvature for Lipschitz gluings and cones

In 2D, the curvature matches Gauss-Bonnet for gluings, differs for cones

Characterization of Ricci flows as super Ricci flows with minimal curvature

Abstract

We present a synthetic notion of scalar curvature (and its integral) for Riemannian manifolds and metric measure spaces, defined in terms of the initial slope of a Gaussian (double) integral. We explicitly calculate the integral scalar curvature for Lipschitz gluings of smooth Riemannian manifolds and for cones. In dimension 2, the former coincides with the formula derived by Gauss-Bonnet, whereas the latter differs. The extension to the time-dependent case allows us to characterize Ricci flows as super Ricci flows with minimal integral curvature functional.