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Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature

Steven Gindi

Abstract

We use Lott's functional and construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature. Consequently, we prove that the blowdown limit is locally an expanding Ricci soliton when the structure group is the three dimensional Heisenberg group. In addition, we classify this soliton when the base manifold is one dimensional. This, together with Lott's work in the abelian setting, yields a complete local classification of invariant Ricci flow blowdown limits on four dimensional, nilpotent principal bundles.

Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature

Abstract

We use Lott's functional and construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature. Consequently, we prove that the blowdown limit is locally an expanding Ricci soliton when the structure group is the three dimensional Heisenberg group. In addition, we classify this soliton when the base manifold is one dimensional. This, together with Lott's work in the abelian setting, yields a complete local classification of invariant Ricci flow blowdown limits on four dimensional, nilpotent principal bundles.

Paper Structure

This paper contains 33 sections, 44 theorems, 127 equations.

Table of Contents

  1. Introduction and Summary of Results
  2. Summary of Results
  3. Background and Notation
  4. Lott's Functional: New Monotonicity Properties and Rigidity Results
  5. Toward a More Complete Functional: Lie Group Case
  6. The $\mathcal{W}_{+}$-Functional for Ricci Flows on Nilpotent Principal Bundles
  7. Acknowledgments
  8. Background: Ricci Flow Blowdown Limits and Expanding Solitons
  9. Ricci Flow Blowdown Limits
  10. Expanding Ricci Solitons
  11. Background: Ricci Curvature and Nilpotent Lie Groups
  12. The $\delta$-map
  13. Ricci Curvature
  14. Ricci Flow on Nilpotent Lie Groups: Solitons and the $\mathcal{W}_{+}$-Functional
  15. Nilsolitons
  16. ...and 18 more sections

Key Result

Theorem 1.3

Let $\overline{g}(t)$ be an invariant Ricci flow defined for $t \in (0, \infty)$ on a nilpotent $\mathcal{G}$-principal bundle $P\rightarrow M$, which is fibered over a compact, $n$-dimensional manifold. Suppose the curvature $F_{\overline{A}(t)}=0$. Also let $f_{t} \in C^{\infty}(M)$ be a solution

Theorems & Definitions (92)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.6
  • Definition 1.7
  • Proposition 1.8
  • Corollary 1.9
  • Definition 1.10
  • Theorem 1.11
  • ...and 82 more