Paper · Sep 4, 2024

Ricci curvature and normalized Ricci flow on generalized Wallach spaces

As on arXiv

Abstract

We proved that the normalized Ricci flow does not preserve the positivity of Ricci curvature of Riemannian metrics on every generalized Wallach space with $a_{1} + a_{2} + a_{3} \leq 1/2$ , in particular on the spaces $SU (k + l + m) / SU (k) \times SU (l) \times SU (m)$ and $Sp (k + l + m) / Sp (k) \times Sp (l) \times Sp (m)$ independently on $k, l$ and $m$ . The positivity of Ricci curvature is preserved for all original metrics with $Ric > 0$ on generalized Wallach spaces $a_{1} + a_{2} + a_{3} > 1/2$ if the conditions $4 (a_{j} + a_{k})^{2} \geq (1 - 2 a_{i}) (1 + 2 a_{i})^{- 1}$ hold for all ${i, j, k} = {1, 2, 3}$ . We also established that the spaces $SO (k + l + m) / SO (k) \times SO (l) \times SO (m)$ satisfy the above conditions for $max {k, l, m} \leq 11$ , moreover, additional conditions were found to keep $Ric > 0$ in cases when $max {k, l, m} \leq 11$ is violated. Similar questions have also been studied for all other generalized Wallach spaces given in the classification of Yuri\uı Nikonorov.

Comments: 35 pages, 13 figures, 5 tables

Ricci flow

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces
Nurlan Abiev2024 PDF

Topics Ricci flow · Same author Nurlan Abiev · Shared terms “generalized Wallach”, “Wallach spaces”, “normalized Ricci”
On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces
Nurlan Abiev2023 PDF

Topics Ricci flow · Same author Nurlan Abiev · Shared terms “Wallach spaces”, “generalized Wallach”, “normalized Ricci”
Refined behavior description of the normalized Ricci flow on homogeneous spaces
Nurlan A. Abiev2025 PDF

Topics Ricci flow · Same author Nurlan Abiev · Shared terms “normalized Ricci”, “generalized Wallach”, “Wallach spaces”
The Ricci curvature and the normalized Ricci flow on the Stiefel manifolds $SO (n) / SO (n - 2)$
Nurlan Abiev2024 PDF

Topics Ricci flow · Same author Nurlan Abiev · Shared terms “normalized Ricci”, “Riemannian metrics”, “positivity”
Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen
David González-Álvaro and Masoumeh Zarei2024 PDF

Topics Ricci flow · Shared terms “Wallach spaces”, “preserved”
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 et al.2023 PDF

Topics Ricci flow · Shared terms “Wallach”
Ricci curvature integrals, local functionals, and the Ricci flow
Yuanqing Ma and Bing Wang2021 PDF

Topics Ricci flow · Shared terms “normalized Ricci”
A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric
Gang Li2025 PDF

Topics Ricci flow · Shared terms “normalized Ricci”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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)

In the guidebook

5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.
11A.1
The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles
A heat equation for the metric: spheres, cylinders and scaling.

Ricci curvature and normalized Ricci flow on generalized Wallach spaces

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters