Paper · Mar 15, 2023

Positive intermediate curvatures and Ricci flow

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Abstract

We show that, for any $n \geq 2$ , there exists a homogeneous space of dimension $d = 8 n - 4$ with metrics of $Ric_{\frac{d}{2} - 5} > 0$ if $n \neq = 3$ and $Ric_{6} > 0$ 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 $n = 2$ .

Comments: Final version

Ricci flow Sphere theorems Scalar curvature Homogeneous spaces

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Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

11A.11
Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles
Positive Ricci curvature in dimension 3 flows to a round metric.
11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.
6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

Positive intermediate curvatures and Ricci flow

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