As on arXiv
Abstract
The avoidance principle says that mean curvature flows of hypersurfaces remain disjoint if they are disjoint at the initial time. We prove several generalizations of the avoidance principle that allow for intersections of hypersurfaces. First, we prove that the Hausdorff dimension of the intersection of two mean curvature flows is non-increasing over time, and we find precise information on how the dimension changes. We then show that the self-intersection of an immersed mean curvature flow has non-increasing dimension over time. Next, we extend the intersection dimension monotonicity to Brakke flows and level set flows which satisfy a localizability condition, and we provide examples showing that the monotonicity fails for general weak solutions. We find a localization result for level set flows with finitely many singularities, and as a consequence, we obtain a fattening criterion for these flows which depends on the behavior of intersections with smooth flows.
Comments: 39 pages, 4 figures. Comments welcome!
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.
-
On the avoidance principle for codimension 1 space-time Brakke flows
Topics Weak & singular flows · Shared terms “avoidance principle”, “Brakke flows”
-
Mean Curvature Flow from Conical Singularities
Topics Weak & singular flows, Singularities · Shared terms “set flow”, “level set”
-
Closed mean curvature flows with asymptotically conical singularities
Topics Weak & singular flows, Singularities · Same author Tang-Kai Lee · Shared terms “set flow”, “level set”, “fattening”
-
Generic regularity of Level Set Flows with spherical singularity
Topics Weak & singular flows, Singularities · Shared terms “set flow”, “level set”
-
On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces
Topics Weak & singular flows, Singularities · Shared terms “set flow”, “level set”, “fattening”
-
Quantitative avoidance for free boundary flows and applications
Topics Weak & singular flows, Singularities · Shared terms “avoidance”, “Brakke flows”
-
Generic level sets in mean curvature flow are BV solutions
Topics Weak & singular flows · Shared terms “level set”, “Brakke flows”
-
Equivalence of weak solution concepts for mean curvature flow
Topics Weak & singular flows · Shared terms “level set”, “Brakke flows”
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
- 12C.9After Perelman
Singular flows, the Smale conjecture, and Bamler’s theory.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
- 6C.8Monotonicity, ε-Regularity and Tangent Cones
Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.