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Ricci flow on Courant algebroids

Jeffrey Streets, Charles Strickland-Constable, Fridrich Valach

Abstract

We develop a theory of Ricci flow for metrics on Courant algebroids which unifies and extends the analytic theory of various geometric flows, yielding a general tool for constructing solutions to supergravity equations. We prove short time existence and uniqueness of solutions on compact manifolds, in turn showing that the Courant isometry group is preserved by the flow. We show a scalar curvature monotonicity formula and prove that generalized Ricci flow is a gradient flow, extending fundamental works of Hamilton and Perelman. Using these we show a convergence result for certain nonsingular solutions to generalized Ricci flow.

Ricci flow on Courant algebroids

Abstract

We develop a theory of Ricci flow for metrics on Courant algebroids which unifies and extends the analytic theory of various geometric flows, yielding a general tool for constructing solutions to supergravity equations. We prove short time existence and uniqueness of solutions on compact manifolds, in turn showing that the Courant isometry group is preserved by the flow. We show a scalar curvature monotonicity formula and prove that generalized Ricci flow is a gradient flow, extending fundamental works of Hamilton and Perelman. Using these we show a convergence result for certain nonsingular solutions to generalized Ricci flow.
Paper Structure (36 sections, 47 theorems, 213 equations, 1 figure)

This paper contains 36 sections, 47 theorems, 213 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Courant algebroids
  3. Definition and first examples
  4. Courant algebroid automorphisms
  5. Courant algebroid connections
  6. Densities
  7. Divergences
  8. Generalized Riemannian geometry
  9. Generalized metrics
  10. Isometries of generalized metrics
  11. Levi-Civita connections
  12. Differential operators
  13. Generalized Riemann and Ricci tensors
  14. Riemann tensor
  15. Ricci tensor
  16. ...and 21 more sections

Key Result

Theorem 1.1

Given $E \to M$ a Courant algebroid over a compact manifold and $(\mathop{\mathrm{\mathcal{G}}}\nolimits_0, \sigma_0)$ a generalized metric on $E$ and half-density on $M$, there exists $\epsilon > 0$ and a unique solution to generalized Ricci flow with initial condition $(\mathop{\mathrm{\mathcal{G}

Figures (1)

  • Figure 1: Construction of $Z_u$

Theorems & Definitions (148)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5: BursztynCavalcantiGualtieri
  • ...and 138 more