Infinite-Time Singularities of the Lagrangian Mean Curvature Flow
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916230298861568 |
|---|---|
| author | Su, Wei-Bo Tsai, Chung-Jun Wood, Albert |
| author_facet | Su, Wei-Bo Tsai, Chung-Jun Wood, Albert |
| contents | In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as $t\to\infty$. In particular, the flow decomposes the initial data into a union of special Lagrangians intersecting at one point. This result shows that infinite-time singularities can form in the Thomas--Yau `semi-stable' situation. A precise polynomial blow-up rate of the second fundamental form is also shown.
The infinite-time singularity formation is obtained by a perturbation of an approximate family $N^{\varepsilon(t)}$ constructed by gluing in special Lagrangian `Lawlor necks' of size $\varepsilon(t)$, where the dynamics of the neck size $\varepsilon(t)$ are driven by the obstruction for the existence of nearby special Lagrangians to $N^{\varepsilon(t)}$. This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02228 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinite-Time Singularities of the Lagrangian Mean Curvature Flow Su, Wei-Bo Tsai, Chung-Jun Wood, Albert Differential Geometry In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as $t\to\infty$. In particular, the flow decomposes the initial data into a union of special Lagrangians intersecting at one point. This result shows that infinite-time singularities can form in the Thomas--Yau `semi-stable' situation. A precise polynomial blow-up rate of the second fundamental form is also shown. The infinite-time singularity formation is obtained by a perturbation of an approximate family $N^{\varepsilon(t)}$ constructed by gluing in special Lagrangian `Lawlor necks' of size $\varepsilon(t)$, where the dynamics of the neck size $\varepsilon(t)$ are driven by the obstruction for the existence of nearby special Lagrangians to $N^{\varepsilon(t)}$. This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow. |
| title | Infinite-Time Singularities of the Lagrangian Mean Curvature Flow |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2401.02228 |