Paper · Jun 6, 2022

Isoparametric hypersurfaces of Riemannian manifolds as initial data for the mean curvature flow

Felippe Guimarães, João Batista Marques dos Santos, João Paulo dos Santos

As on arXiv

Abstract

We show that the evolution of isoparametric hypersurfaces of Riemannian manifolds by the mean curvature flow is given by a reparametrization of the parallel family in short time, as long as the uniqueness of the mean curvature flow holds for the initial data and the corresponding ambient space. As an application, we provide a class of Riemannian manifolds that admit hypersurfaces with constant principal curvatures, which are not isoparametric hypersurfaces. Furthermore, for a class of ambient spaces, we show that the singularities developed by the mean curvature flow with isoparametric hypersurfaces as the initial data are Type I singularities. We apply our results to describe the evolution of isoparametric hypersurfaces by the mean curvature flow in ambient spaces with nonconstant sectional curvature, such as homogenous 3-manifolds $E (κ, τ)$ with 4-dimensional isometry groups, and Riemannian products $Q_{c_{1}}^{2} \times Q_{c_{2}}^{2}$ of space forms.

Comments: References on submanifolds with higher codimensions were included in this version

Mean curvature flow Singularities 3-manifolds 4-manifolds Sphere theorems Stability & uniqueness

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

Solitons to Mean Curvature Flow in the hyperbolic 3-space
R. F. de Lima, A. K. Ramos and J. P. dos Santos2023Wider flows PDF

Topics Stability & uniqueness · Same authors João Batista Marques dos Santos, João Paulo dos Santos
Uniqueness of the $[φ, e_{3}]$ -catenary cylinders by their asymptotic behaviour
A. L. Martínez-Triviño and J. P. dos Santos2021Wider flows PDF

Topics Stability & uniqueness · Same authors João Batista Marques dos Santos, João Paulo dos Santos
Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in $H^{3}$ as Translation Surfaces
Tarcios Andrey Ferreira and João Paulo dos Santos2025Wider flows PDF

Topics Mean curvature flow · Same authors João Batista Marques dos Santos, João Paulo dos Santos
Local structure of gradient almost Ricci solitons with harmonic Weyl tensor
Valter Borges, Matheus Andrade Ribeiro de Moura Horácio and João Paulo dos Santos2025 PDF

Topics 4-manifolds · Same authors João Batista Marques dos Santos, João Paulo dos Santos
Rotators-Translators to Mean Curvature Flow in $H^{2} \times R$
Ronaldo F. de Lima, Álvaro K. Ramos and João Paulo dos Santos2024Wider flows PDF

Topics Mean curvature flow · Same authors João Batista Marques dos Santos, João Paulo dos Santos
On the inverse mean curvature flow by parallel hypersurfaces in space forms
Alancoc dos Santos Alencar and Keti Tenenblat2024Wider flows PDF

Topics Sphere theorems · Shared terms “isoparametric”, “space forms”, “principal curvatures”
Contraction of hypersurfaces with positive sectional curvature in hyperbolic space
Tianci Luo, Yong Wei and Rong Zhou2026Wider flows PDF

Topics Sphere theorems, Singularities · Shared terms “sectional curvature”, “principal curvatures”
The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor
Valter Borges, Matheus Andrade Ribeiro de Moura Horácio and João Paulo dos Santos2025 PDF

Same authors João Batista Marques dos Santos, João Paulo dos Santos

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

In the guidebook

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.
10A.1
A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.
10A.2
Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization
Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

Isoparametric hypersurfaces of Riemannian manifolds as initial data for the mean curvature flow

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters