Paper · Dec 4, 2024
The Ricci curvature and the normalized Ricci flow on the Stiefel manifolds
As on arXiv
Abstract
We proved that on every Stiefel manifold with the normalized Ricci flow preserves the positivity of the Ricci curvature of invariant Riemannian metrics with positive Ricci curvature. Moreover, the normalized Ricci flow evolves all metrics with mixed Ricci curvature into metrics with positive Ricci curvature in finite time. From the point of view of the theory of dynamical systems we proved that for every invariant set of the normalized Ricci flow on defined as , , there exists a smaller invariant set for every , where is the domain in responsible for parameters of invariant Riemannian metrics on admitting positive Ricci curvature.
Comments: 15 pages, 4 figures
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
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Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
- 11B.6Rotationally Symmetric Flows and the Standard Solution
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- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.