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1. 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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2. Regularity of the Level Set Flow

   https://doi.org/10.1002/cpa.21703  

   Colding, Tobias Holck; Minicozzi, William P (July 2017, Communications on Pure and Applied Mathematics)

   Abstract We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative and satisfies the equation classically. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closedC¹manifold with cylindrical singularities. © 2017 Wiley Periodicals, Inc. 

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3. Uniqueness of Regular Tangent Cones for Immersed Stable Hypersurfaces

   https://doi.org/10.1007/s00205-024-02071-y  

   Edelen, Nick; Minter, Paul (November 2024, Archive for Rational Mechanics and Analysis)

   Abstract We establish uniqueness and regularity results for tangent cones (at a point or at infinity), with isolated singularities arising from a given immersed stable minimal hypersurface with suitably small (non-immersed) singular set. In particular, our results allow the tangent cone to occur with any integer multiplicity. 

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4. Deformations of Calabi–Yau Varieties With Isolated Log Canonical Singularities

   https://doi.org/10.1093/imrn/rnaf112  

   Friedman, Robert; Laza, Radu (May 2025, International Mathematics Research Notices)

   Abstract Recent progress in the deformation theory of Calabi–Yau varieties $$Y$$ with canonical singularities has highlighted the key role played by the higher Du Bois and higher rational singularities, and especially by the so-called $$k$$-liminal singularities for $$k\ge 1$$. The goal of this paper is to show that certain aspects of this study extend naturally to the $$0$$-liminal case as well, that is, to Calabi–Yau varieties $$Y$$ with Gorenstein log canonical, but not canonical, singularities. In particular, we show the existence of first order smoothings of $$Y$$ in the case of isolated $$0$$-liminal hypersurface singularities, and extend Namikawa’s unobstructedness theorem for deformations of singular Calabi–Yau three-folds $$Y$$ with canonical singularities to the case where $$Y$$ has an isolated $$0$$-liminal lci singularity under suitable hypotheses. Finally, we describe an interesting series of examples. 

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5. Exact Lagrangian tori in symplectic Milnor fibers constructed with fillings

   https://doi.org/10.2140/agt.2025.25.3225  

   Capovilla-Searle, Orsola (January 2025, Algebraic & Geometric Topology)

   We use exact Lagrangian fillings and Weinstein handlebody diagrams to construct infinitely many distinct exact Lagrangian tori in 4-dimensional Milnor fibers of isolated hypersurface singularities with positive modality. We also provide a generalization of a criterion for when the symplectic homology of a Weinstein 4-manifold is nonvanishing given an explicit Weinstein handlebody diagram. 

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