Paper · Jul 12, 2023

The complete dynamics description of positively curved metrics in the Wallach flag manifold $SU (3) / T^{2}$

Leonardo F. Cavenaghi, Lino Grama, Ricardo M. Martins, Douglas D. Novaes

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Abstract

The family of invariant Riemannian manifolds in the Wallach flag manifold $SU (3) / T^{2}$ is described by three parameters $(x, y, z)$ of positive real numbers. By restricting such a family of metrics in the tetrahedron $T := x + y + z = 1$ , in this paper, we describe all regions $R \subset T$ admitting metrics with curvature properties varying from positive sectional curvature to positive scalar curvature, including positive intermediate curvature notion's. We study the dynamics of such regions under the projected Ricci flow in the plane $(x, y)$ , concluding sign curvature maintenance and escaping. In addition, we obtain some results for positive intermediate Ricci curvature for a path of metrics on fiber bundles over $SU (3) / T^{2}$ , further studying its evolution under the Ricci flow on the base.

Comments: 15 pages, 12 figures. arXiv admin note: text overlap with arXiv:2305.06119

Ricci flow Sphere theorems Scalar curvature

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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)

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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.
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.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

The complete dynamics description of positively curved metrics in the Wallach flag manifold $SU (3) / T^{2}$

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The complete dynamics description of positively curved metrics in the Wallach flag manifold SU(3)/T2

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The complete dynamics description of positively curved metrics in the Wallach flag manifold $SU (3) / T^{2}$