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

Marco Flaim, Erik Hupp, Karl-Theodor Sturm

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.

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

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.
Paper Structure (45 sections, 38 theorems, 231 equations, 6 figures)

This paper contains 45 sections, 38 theorems, 231 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The Gaussian volume functional and (integral) scalar curvature
  4. The Gaussian volume functional and the Gaussian double integral
  5. Assumption:
  6. Scalar curvature
  7. The directional Gaussian volume
  8. Examples of Gaussian volume functional
  9. Weighted Riemannian manifolds
  10. RCD Spaces
  11. Manifolds with boundary, doubling, tripling etc.
  12. Manifolds with boundary
  13. Doubling
  14. Gluing multiple copies
  15. Lipschitz gluing of manifolds
  16. ...and 30 more sections

Key Result

Lemma 2.2

Assume $(X,d,m)$ satisfies density. Then (i) The limit ${\mathfrak a}_0(x):= \lim_{s\searrow0}{\mathfrak a}_s(x)$ exists and coincides with $\rho(x)$. (ii) Similarly, when $m$ is finite, ${\mathfrak A}_0:=\lim_{s\searrow0}{\mathfrak A}_s=\int_X\rho\,dm$. (iii) Moreover,

Figures (6)

  • Figure 1: Graph of $\mathfrak C$.
  • Figure 2: Radial $d$-geodesics ending in the regions $W_x$, $V_x$, and $E_x$.
  • Figure 3: When $\alpha < 0$, $R>r$, $\partial B_R = \partial B_R^{(1)} \sqcup \partial B_R^{(2)}$
  • Figure 4: $\alpha\in[0,\pi]$, $R>r$
  • Figure 5: $\alpha\in[\pi,2\pi)$, $R>r$
  • ...and 1 more figures

Theorems & Definitions (91)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • proof
  • Theorem 2.7
  • proof : Proof I (expanding the volume of spheres)
  • ...and 81 more