Paper · Jan 4, 2024
Infinite-Time Singularities of the Lagrangian Mean Curvature Flow
As on arXiv
Abstract
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 . 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 constructed by gluing in special Lagrangian `Lawlor necks' of size , where the dynamics of the neck size are driven by the obstruction for the existence of nearby special Lagrangians to . This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow.
Comments: 52 pages; update the references; simplify section 4 and 5
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.
-
Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces
Topics Lagrangian mean curvature flow, Singularities · Same author Chung-Jun Tsai · Shared terms “infinite time”, “time singularity”, “special Lagrangian”
-
Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics
Topics Lagrangian mean curvature flow, Singularities · Same author Wei-Bo Su · Shared terms “special Lagrangian”, “time singularity”, “initial data”
-
Singularity formations in Lagrangian mean curvature flow
Topics Lagrangian mean curvature flow, Singularities · Shared terms “special Lagrangian”, “Lawlor necks”, “singularity formation”
-
Cohomogeneity-One Lagrangian Mean Curvature Flow
Topics Lagrangian mean curvature flow, Singularities · Same author Albert Wood · Shared terms “special Lagrangian”, “time singularity”
-
Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow
Topics Lagrangian mean curvature flow, Singularities · Same author Wei-Bo Su · Shared terms “special Lagrangian”, “time singularity”
-
Entire solutions of two-convex Lagrangian mean curvature flows
Topics Lagrangian mean curvature flow · Same author Chung-Jun Tsai · Shared terms “special Lagrangian”
-
Ancient solutions and translators of Lagrangian mean curvature flow
Topics Lagrangian mean curvature flow, Ancient solutions · Shared terms “special Lagrangian”
-
Mean Curvature Flows of Two-Convex Lagrangians
Topics Lagrangian mean curvature flow · Same author Chung-Jun Tsai
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.
In the guidebook
- 12B.3The Structure of κ-Solutions
Necks and caps, compactness, and universal estimates.
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
- 11B.6Rotationally Symmetric Flows and the Standard Solution
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.