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Positive intermediate curvatures and Ricci flow

David González-Álvaro, Masoumeh Zarei

TL;DR

The paper addresses whether intermediate curvature conditions $\mathrm{Ric}_k>0$ are preserved by the Ricci flow. It constructs $\mathsf{Sp}_{n+1}$-invariant metrics on the homogeneous spaces $P_n$ of dimension $d=8n-4$ with $\mathrm{Ric}_{k(n)}>0$ that, under normalized Ricci flow, evolve to metrics whose Ricci tensor has negative eigenvalues, i.e. is not $(d-4)$-positive. The method extends Böhm–Wilking’s $n=2$ result to the full family of spaces $P_n$ by reducing the flow to a two-variable ODE system on invariant submersion metrics and employing a coordinate change to drive certain Ricci-eigenvalues negative. This demonstrates that wide ranges of curvature-positivity conditions are not invariant under Ricci flow, clarifying the limits of curvature-structure preservation in higher dimensions.

Abstract

We show that, for any $n\geq 2$, there exists a homogeneous space of dimension $d=8n-4$ with metrics of $\mathrm{Ric}_{\frac{d}{2}-5}>0$ if $n\neq 3$ and $\mathrm{Ric}_6>0$ if $n=3$ which evolve under the Ricci flow to metrics whose Ricci tensor is not $(d-4)$-positive. Consequently, Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. This extends a theorem of Böhm and Wilking in the case of $n=2$.

Positive intermediate curvatures and Ricci flow

TL;DR

The paper addresses whether intermediate curvature conditions are preserved by the Ricci flow. It constructs -invariant metrics on the homogeneous spaces of dimension with that, under normalized Ricci flow, evolve to metrics whose Ricci tensor has negative eigenvalues, i.e. is not -positive. The method extends Böhm–Wilking’s result to the full family of spaces by reducing the flow to a two-variable ODE system on invariant submersion metrics and employing a coordinate change to drive certain Ricci-eigenvalues negative. This demonstrates that wide ranges of curvature-positivity conditions are not invariant under Ricci flow, clarifying the limits of curvature-structure preservation in higher dimensions.

Abstract

We show that, for any , there exists a homogeneous space of dimension with metrics of if and if which evolve under the Ricci flow to metrics whose Ricci tensor is not -positive. Consequently, Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. This extends a theorem of Böhm and Wilking in the case of .
Paper Structure (6 sections, 3 theorems, 32 equations)

This paper contains 6 sections, 3 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['T:Main']}
  3. Preliminaries
  4. The metrics of $\mathop{\mathrm{Ric}}\nolimits_{k(n)}>0$
  5. Normalized Ricci flow
  6. Proof of Theorem \ref{['T:Main']}

Key Result

Theorem 1

For any $n\geq 2$ there exists a homogeneous space of dimension $d=8n-4$ with metrics of $\mathop{\mathrm{Ric}}\nolimits_{\frac{d}{2}-5}>0$ if $n\neq 3$ and $\mathop{\mathrm{Ric}}\nolimits_6>0$ if $n=3$ which evolve under the Ricci flow to metrics whose Ricci tensor is not $(d-4)$-positive.

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 2
  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3