Strong spherical rigidity of ancient solutions of expansive curvature flows

Risa, Susanna; Sinestrari, Carlo

doi:10.1112/blms.12308

Cited by 10 publications

(5 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, we want to mention that after the first version of this paper appeared on arXiv.org in 2016, many other results on ancient solutions to curvature flows appeared, see [5,24] for a classification of ancient solutions using Aleksandrov reflection and [6,7,19,20] for other related results.…”

Section: Introductionmentioning

confidence: 99%

On the classification of ancient solutions to curvature flows on the sphere

Bryan¹,

Ivaki²,

Scheuer³

2024

Annali Scuola Normale Superiore - Classe Di Scienze

View full text Add to dashboard Cite

We consider the evolution of hypersurfaces on the unit sphere S n+1 by smooth functions of the Weingarten map. We introduce the notion of 'quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Aleksandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases.

show abstract

Section: Introductionmentioning

confidence: 99%

On the classification of ancient solutions to curvature flows on the sphere

Bryan¹,

Ivaki²,

Scheuer³

2024

Annali Scuola Normale Superiore - Classe Di Scienze

View full text Add to dashboard Cite

show abstract

“…This suggests that the existence of ovaloids is a quite general feature of curvature flows and is not related to particular properties of the speed, except possibly for some growth constraint. On the other hand, it is interesting to remark that there are geometric flows in different settings where no analogue of the ovaloids exists: this has been shown for flows in the sphere in [13] and for expanding flows in Euclidean space in [29].…”

Section: Introductionmentioning

confidence: 99%

Non-homothetic convex ancient solutions for flows by high powers of curvature

Risa¹,

Sinestrari²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We prove the existence of closed convex ancient solutions to curvature flows which become more and more oval for large negative times. The speed function is a general symmetric function of the principal curvatures, homogeneous of degree greater than one. This generalises previous work on the mean curvature flow and other one-homogeneous curvature flows. As an auxiliary result, we prove a new theorem on the convergence to a round point of convex rotationally symmetric hypersurfaces satisfying a suitable constraint on the curvatures.

show abstract

“…Different types of speed have been studied and many different problems addressed. Just to mention some of the most recent results, Sinestrari produced convexity estimates with Alessandroni [1] and considered volume and area preserving flows with Bertini [2] and ancient solutions for a very general class of expanding flows with Risa [3]. McCoy considered contracting nonhomogeneous flows [4] and their self-similar solutions [5].…”

Section: Introductionmentioning

confidence: 99%

Nonhomogeneous expanding flows in hyperbolic spaces

Pipoli

2022

Ann Glob Anal Geom

View full text Add to dashboard Cite

In the present paper, we consider star-shaped mean convex hypersurfaces of the real, complex and quaternionic hyperbolic space evolving by a class of nonhomogeneous expanding flows. For any choice of the ambient manifold, the initial conditions are preserved and the long-time existence of the flow is proved. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling, the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space; otherwise, it converges to a conformal multiple of the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in every case, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.

show abstract

Strong spherical rigidity of ancient solutions of expansive curvature flows

Cited by 10 publications

References 34 publications

On the classification of ancient solutions to curvature flows on the sphere

On the classification of ancient solutions to curvature flows on the sphere

Non-homothetic convex ancient solutions for flows by high powers of curvature

Nonhomogeneous expanding flows in hyperbolic spaces

Contact Info

Product

Resources

About