Classification of ancient ovals in higher dimensional mean curvature flow
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908765100441600 |
|---|---|
| author | Choi, Beomjun Du, Wenkui Zhao, Ziyi |
| author_facet | Choi, Beomjun Du, Wenkui Zhao, Ziyi |
| contents | We study compact non-selfsimilar ancient noncollapsed solutions to the mean curvature flow in $\mathbb{R}^{n+1}$, called ancient ovals. Our main result is the classification of $k$-ovals: any $k$-oval (characterized by having cylindrical blow down $\mathbb{R}^k\times S^{n-k}$ and the quadratic bending asymptotics) belongs, up to space-time rigid motions and parabolic dilations, to the family of ancient ovals constructed by Haslhofer and the second author. Assuming the nonexistence of exotic ovals (recently proved by Bamler-Lai), this yields a classification of all ancient ovals and identifies the moduli space, modulo symmetries, with an open $(k-1)$-simplex modulo the symmetry of simplex. Although these conclusions are contained in the recent breakthrough of Bamler-Lai classifying all ancient asymptotically cylindrical flows and resolving the mean convex neighborhood conjecture, we give an alternative argument for the independently obtained classification of $k$-ovals in arbitrary dimensions based on a different spectral parametrization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09441 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classification of ancient ovals in higher dimensional mean curvature flow Choi, Beomjun Du, Wenkui Zhao, Ziyi Differential Geometry Analysis of PDEs We study compact non-selfsimilar ancient noncollapsed solutions to the mean curvature flow in $\mathbb{R}^{n+1}$, called ancient ovals. Our main result is the classification of $k$-ovals: any $k$-oval (characterized by having cylindrical blow down $\mathbb{R}^k\times S^{n-k}$ and the quadratic bending asymptotics) belongs, up to space-time rigid motions and parabolic dilations, to the family of ancient ovals constructed by Haslhofer and the second author. Assuming the nonexistence of exotic ovals (recently proved by Bamler-Lai), this yields a classification of all ancient ovals and identifies the moduli space, modulo symmetries, with an open $(k-1)$-simplex modulo the symmetry of simplex. Although these conclusions are contained in the recent breakthrough of Bamler-Lai classifying all ancient asymptotically cylindrical flows and resolving the mean convex neighborhood conjecture, we give an alternative argument for the independently obtained classification of $k$-ovals in arbitrary dimensions based on a different spectral parametrization. |
| title | Classification of ancient ovals in higher dimensional mean curvature flow |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2601.09441 |