Ricci flow on certain homogeneous spaces

Statha, Marina

doi:10.1007/s10455-022-09843-3

Cited by 3 publications

(2 citation statements)

References 31 publications

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“…On the other hand, the Ricci flow is feasible to handle (due to betterestablished techniques) in the homogeneous setting, [15,25,14,13,26]. Of particular interest is the recent paper [15], substantiated in the following: It consists in appropriately normalizing the Ricci flow to a simplex followed by a time reparametrization to obtain polynomial equations, leading to a projected Ricci flow.…”

Section: Introductionmentioning

confidence: 99%

On the dynamics of positively curved metrics on SU(3)/T 2 under the homogeneous Ricci flow

Cavenaghi,

Grama,

Martins

2024

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In this note, we show that the classical Wallach manifold SU(3)/T 2admits metrics of positive intermediate Ricci curvature (Ric d > 0) for d = 1, 2, 3, 4, 5 that lose these properties under the homogeneous Ricci flow for d = 1, 2, 3, 5. We make the same analyses to the family of Riemannian flag manifolds SU(m + 2p)/S(U(m) × U(p) × U(p)), concluding similar results. These explicitly verify some claims expected to be true among experts (see [7]) for positive Ricci curvature and intermediate positive Ricci curvature. Our technique is only possible due to the global behavior understanding of the homogeneous Ricci flow for invariant metrics on these manifolds.

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Section: Introductionmentioning

confidence: 99%

On the dynamics of positively curved metrics on SU(3)/T 2 under the homogeneous Ricci flow

Cavenaghi,

Grama,

Martins

2024

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“…Exploring the Ricci flow on homogeneous spaces involves tools from the theory of dynamical systems. This approach has been adopted by various works covering different classes of homogeneous spaces, including generalized Wallach spaces [1,23], Stiefel manifolds [23], and complex flag manifolds [13,14,23].…”

Section: Introductionmentioning

confidence: 99%

Geodesic orbit spaces in real flag manifolds

Brian

Grama

Negreiros

2020

Communications in Analysis and Geometry

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In this paper, we investigate homogeneous Riemannian geometry on real flag manifolds of the split real form of g 2 . We characterize the metrics that are invariant under the action of a maximal compact subgroup of G 2 . Our exploration encompasses the analysis of g.o. metrics and equigeodesics on the g 2 -type flag manifolds. Additionally, we explore the Ricci flow for the case where the isotropy representation has no equivalent summands, employing techniques from the qualitative theory of dynamical systems.

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Positive intermediate curvatures and Ricci flow

González-Álvaro,

Zarei

2024

Proc. Amer. Math. Soc.

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We show that, for any n ≥ 2 n\geq 2 , there exists a homogeneous space of dimension d = 8 n − 4 d=8n-4 with metrics of R i c d 2 − 5 > 0 Ric_{\frac {d}{2}-5}>0 if n ≠ 3 n\neq 3 and R i c 6 > 0 Ric_6>0 if n = 3 n=3 which evolve under the Ricci flow to metrics whose Ricci tensor is not ( d − 4 ) (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 [Geom. Funct. Anal. 17 (2007), pp. 665–681] in the case of n = 2 n=2 .

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Ricci flow on certain homogeneous spaces

Cited by 3 publications

References 31 publications

On the dynamics of positively curved metrics on SU(3)/T 2 under the homogeneous Ricci flow

On the dynamics of positively curved metrics on SU(3)/T 2 under the homogeneous Ricci flow

Geodesic orbit spaces in real flag manifolds

Positive intermediate curvatures and Ricci flow

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