An Intersection Principle for Mean Curvature Flow

TL;DR

The paper develops an intersection principle for mean curvature flow, showing that the codimension-two Hausdorff measure and the dimension of the intersection of two flows is non-increasing in time under suitable smooth or weak-solution settings. It introduces localizability and nonfattening conditions to extend the principle from smooth flows to Brakke and level set flows, and provides precise finiteness and one-sidedness tools (via Huang–Jiang bounds and White’s topological results). The work connects classical avoidance to higher-codimension intersections, yields a self-intersection theory for immersed MCFs, and furnishes fattening criteria and several counterexamples clarifying the limits of monotonicity in weak solutions. The results have implications for uniqueness through singular times, for localization of LSFs, and for understanding when intersection behavior can control flow geometry or signal fattening phenomena.

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.

Figures (4)

• Figure 1: A flow ${\mathcal{M}}_t$ with an isolated conical singularity can intersect a plane $P=P_t$ on a $0$-dimensional set at the singular time $t=0$ yet intersect on a higher-dimensional set at a later time $t=\varepsilon$.
• Figure 2: One-sidedness may fail for level set flows intersecting on a small set. In this case, one of the LSFs may be localizable, meaning that it decomposes as a flow of subsets. This figure shows an LSF that could evolve as the union of the flow starting from the portion inside the sphere and another starting from the portion outside the sphere.
• Figure 3: An example of a profile curve satisfying the desired properties. The curve is far away from the rotation axis so that the principal curvatures in those directions are small.
• Figure 4: The blue curve is an example of a profile curve satisfying the desired properties. The surface obtained by rotating the blue curve contains two round spheres $S_1$ and $S_2$ in the enclosed region, and its neck part can be surrounded by a shrinking torus. In a short time, the number of components of its intersection with the plane, obtained by rotating the straight red line around the $y$-axis, will increase from one to two.

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: HJ
• Proposition 2.3
• proof
• Theorem 2.10
• Remark 2.11
• proof
• Lemma 2.17: HJ