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On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces
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abstract
We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in $\mathbb{R}^3$. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining our work with results of Brendle and Choi-Haslhofer-Hershkovits-White, we show that any level set flow starting from an embedded surface diffeomorphic to a 2-spheres does not fatten. In fact, we obtain that the problem of evolving embedded 2-spheres via the mean curvature flow equation is well-posed within a natural class of singular solutions. Second, we use our result to remove an additional condition in recent work of Chodosh-Choi-Mantoulidis-Schulze. This shows that mean curvature flows starting from any generic embedded surface only incur cylindrical or spherical singularities. Third, our approach offers a new regularity theory for solutions of mean curvature flows that flow through singularities. Among other things, this theory also applies to the innermost and outermost flow of any embedded surface and shows that all singularity models of such flows must have multiplicity one. It also establishes equality of the fattening time with the discrepancy time. Lastly, we obtain a number of further results characterizing a separation phenomenon of mean curvature flows of surfaces.
Forward citations
Cited by 6 Pith papers
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An Ancient Stacked Pancake Solution to Mean Curvature Flow
For every dimension n≥3, there exists an embedded, rotationally symmetric, non-convex ancient mean curvature flow that looks like two parallel pancakes joined by a neck and lies in a slab.
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Singularities of mean curvature flow with bounded mean curvature and Morse index
For closed smooth mean curvature flows in R^{n+1}, 3≤n≤6, at the first singular time either the mean curvature or the Morse index must blow up.
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Classification of ancient noncollapsed flows in $\mathbb{R}^4$
Every ancient noncollapsed mean curvature flow in R^4 is one of the known shrinkers, bowls, ovals, or Hoffman-Ilmanen-Martin-White translators.
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Regularity of cylindrical singular sets of mean curvature flow
Degenerate cylindrical singular sets of mean curvature flow are locally contained in C^{2,α} submanifolds, with curvature determined by the flow's asymptotic profile.
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Passing through nondegenerate singularities in mean curvature flows
Mean curvature flows through nondegenerate cylindrical singularities undergo an isolated, graphical surgery event whose topology change equals an (n-k)-surgery, matching Morse level sets.
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When can minimal hypersurfaces be connected by mean curvature flow?
Min-max minimal spheres in infinitely many Berger 3-spheres are never connected by eternal mean curvature flow to lower-area minimal surfaces, due to a genus obstruction.
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