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1. Mean curvature flow with generic initial data

   https://doi.org/10.1007/s00222-024-01258-0  

   Chodosh, Otis; Choi, Kyeongsu; Mantoulidis, Christos; Schulze, Felix (April 2024, Inventiones mathematicae)

   Abstract We show that the mean curvature flow of generic closed surfaces in$$\mathbb{R}^{3}$$ $R^3$avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in$$\mathbb{R}^{4}$$ $R^4$is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons. 

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2. Differentiability of the Arrival Time

   https://doi.org/10.1002/cpa.21635  

   Colding, Tobias Holck; Minicozzi, William P (December 2016, Communications on Pure and Applied Mathematics)

   Abstract For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite codimensional 2 Hausdorff measure. For a monotonically advancing front, the arrival time is equivalent to the level set method, a~priori not even differentiable but only satisfying the equation in the viscosity sense . Using that it is twice differentiable and that we can identify the Hessian at critical points, we show that it satisfies the equation in the classical sense. The arrival time has a game theoretic interpretation. For the linear heat equation, there is a game theoretic interpretation that relates to Black‐Scholes option pricing. From variations of the Sard and Łojasiewicz theorems, we relate differentiability to whether singularities all occur at only finitely many times for flows.© 2016 Wiley Periodicals, Inc. 

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3. Local tropicalizations of splice type surface singularities

   Cueto, Maria Angelica; Popescu-Pampu, Patrick; Stepanov, Dmitry; Wahl, Jonathan (December 2023, Mathematische Annalen)

   Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham–Brieskorn–Hamm complete intersections of dimension two. Their construction depends on a weighted tree called a splice diagram. In this paper, we study these singularities from the tropical viewpoint. We characterize their local tropicalizations as the cones over the appropriately embedded associated splice diagrams. As a corollary, we reprove some of Neumann and Wahl’s earlier results on these singularities by purely tropical methods, and show that splice type surface singularities are Newton non-degenerate complete intersections in the sense of Khovanskii. We also confirm that under suitable coprimality conditions on its weights, the diagram can be uniquely recovered from the local tropicalization. As a corollary of the Newton non-degeneracy property, we obtain an alternative proof of a recent theorem of de Felipe, González Pérez and Mourtada, stating that embedded resolutions of any plane curve singularity can be achieved by a single toric morphism, after re-embedding the ambient smooth surface germ in a higher-dimensional smooth space. The paper ends with an appendix by Jonathan Wahl, proving a criterion of regularity of a sequence in a ring of convergent power series, given the regularity of an associated sequence of initial forms. 

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4. Mean curvature flow from conical singularities

   https://doi.org/10.1007/s00222-024-01296-8  

   Chodosh, Otis; Daniels-Holgate, J M; Schulze, Felix (December 2024, Inventiones mathematicae)

   Abstract We prove Ilmanen’s resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface$$M^{n}\subset \mathbb{R}^{n+1}$$ $M^n\subset R^{n+1}$,$$2\leq n\leq 6$$ $2\le n\le 6$, with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens. 

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5. Mean curvature flow

   https://doi.org/10.1090/S0273-0979-2015-01468-0  

   Colding, Tobias; Minicozzi, William; Pedersen, Erik (January 2015, Bulletin of the American Mathematical Society)

   Mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. If the hypersurface is in general or generic position, then we explain what singularities can occur under the flow, what the flow looks like near these singularities, and what this implies for the structure of the singular set. At the end, we will briefly discuss how one may be able to use the flow in low-dimensional topology. 

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