Ancient solutions to mean curvature flow for isoparametric submanifolds
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Mathematische Annalen
Ancient solutions to mean curvature flow for isoparametric submanifolds Xiaobo Liu1 · Chuu-Lian Terng2 Received: 20 January 2020 / Revised: 27 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied in Liu and Terng (Duke Math J 147(1):157–179, 2009). In this paper, we will show that all these solutions are ancient solutions and study their limits as time goes to negative infinity. We also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the Chern’s conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres.
1 Introduction The mean curvature flow (abbreviated as MCF) of a submanifold M in a Riemannian manifold X over an interval I is a map f : I × M −→ X satisfying ∂f = H (t, ·), ∂t where H (t, ·) is the mean curvature vector field of f (t, ·). If a solution to this equation exists for all t ∈ (−∞, T ) for some T ≥ 0, then it is called an ancient solution. Ancient solutions are important in studying singularities of MCF. A simple example of ancient solution to MCF is the shrinking sphere in a Euclidean space. A set of conditions
Communicated by F.C. Marques. X. Liu: Research was partially supported by NSFC Grants 11890662, 11890660, and 11431001.
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Xiaobo Liu [email protected] Chuu-Lian Terng [email protected]
1
Beijing International Center for Mathematical Research and School of Mathematical Sciences, Peking University, Beijing, China
2
Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, USA
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X. Liu
which ensure a compact ancient solution to be the shrinking sphere is given in [16]. Other examples of compact convex ancient solutions for MCF of hypersurfaces in Euclidean spaces can be found in [5,7,14,32,33], etc. Recently an ancient solution of MCF of hypersurfaces with the topology of S 1 × S n−1 in Rn+1 was given in [6]. A construction of higher codimensional curve shortening flows was given in [3] and [29]. It was proved in [26] that after reparametrization the family of proper Dupin submanifolds in sphere constructed in [25] is an ancient solution for the MCF of submanifolds in spheres. In this paper, we will give a class of ancient solutions to MCF in Euclidean spaces and spheres for compact submanifolds. These examples include both hypersurfaces and higher codimensional submanifolds in spheres and have more complicated topological types. A submanifold M of a space form is isoparametric if its normal bundle is flat and principal curvatures along any parallel normal vector field are constant. The following results were proved in [30]: (i) If M is a compact isoparametric submanifold in Rn+k , then M is contained in a hypersphere. (ii) The set of parallel submanifolds to M forms a singular foliation, whose top dimensional leaves are also isoparametric and lower dimensional leaves are smooth focal submanifolds of M. Focal submanifolds are
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