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Ancient and eternal solutions to mean curvature flow from minimal surfaces

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Abstract

We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in \({\mathbb {R}}^{n+1}\) for \(n \ge 2\). These provide examples of mean convex yet nonconvex ancient solutions that are not solitons, meaning that they do not evolve by rigid motions or homotheties. Moreover, we construct embedded eternal solutions to mean curvature flow in \({\mathbb {R}}^{n+1}\) for \(n \ge 2\). These eternal solutions are not solitons, are \(O(n)\times O(1)\)-invariant, and are mean convex yet nonconvex. They flow out of the catenoid and are the rotation of a profile curve which becomes infinitely far from the axis of rotation. As \(t \rightarrow \infty \), the profile curves converge to a grim reaper for \(n \ge 3\) and become flat for \(n=2\). Concerning these eternal solutions, we also show they are asymptotically unique up to scale among the embedded \(O(n)\times O(1)\)-invariant, eternal solutions with uniformly bounded curvature and a sign on mean curvature.

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Notes

  1. In line with [18], we mean than on some bounded domain \(D \subset M\), \(\lambda _1(D) < 0\) for the Jacobi operator.

  2. This is with respect to the normal on \(M^1_t\) compatible, as \(t \rightarrow -\infty \), with the normal \(\nu \) to the catenoid \(M^1\) (see the discussion above the theorem).

  3. Technically, \(M^1_t\) will be a double-sheeted graph over B(x, R) by considering the part of \(u_t|_{x<0}\) lying over B(x, R), but this does not affect the application of Theorem 2.3, which will only be applied to the part of \(M^1_t\) corresponding to \(u_t|_{x>0}\).

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Acknowledgements

The authors would like to thank Kyeongsu Choi and Christos Mantoulidis for responding to comments and writing an interesting and inspiring paper [12] where they construct ancient flows out of compact minimal hypersurfaces in non-Euclidean ambient spaces, using different methods than those of this paper. The authors would also like to thank Mat Langford and Shengwen Wang for their comments and suggestions, as well as the referee for their suggested edits. Finally, the authors thank their advisors, Richard Schoen and Bruce Kleiner, respectively, for their support and advice.

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  1. Department of Mathematics, University of California Irvine, Irvine, CA, 92617, United States of America

    Alexander Mramor

  2. Courant Institute, New York University, New York, NY, 10012, United States of America

    Alec Payne

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  1. Alexander Mramor
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Correspondence to Alec Payne.

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Communicated by F. C. Marques.

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Mramor, A., Payne, A. Ancient and eternal solutions to mean curvature flow from minimal surfaces. Math. Ann. 380, 569–591 (2021). https://doi.org/10.1007/s00208-021-02149-y

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