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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. 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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3. Local Well-Posedness of Skew Mean Curvature Flow for Small Data in $$d\ge 4$$ Dimensions

   https://doi.org/10.1007/s00220-021-04303-8  

   Huang, Jiaxi; Tataru, Daniel (January 2022, Communications in Mathematical Physics)

   Abstract The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${{\mathbb {R}}}^{d+2}$$ $R^{d+2}$(or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\ge 4$$ $d\ge 4$. 

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4. Ricci flow does not preserve positive sectional curvature in dimension four

   https://doi.org/10.1007/s00526-022-02335-z  

   Bettiol, Renato G.; Krishnan, Anusha M. (November 2022, Calculus of Variations and Partial Differential Equations)

   Abstract We find examples of cohomogeneity one metrics on$$S^4$$ $S^4$and$$\mathbb {C}P^2$$ $C{P}^2$with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow. 

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5. Curvature and sharp growth rates of log-quasimodes on compact manifolds

   https://doi.org/10.1007/s00222-025-01315-2  

   Huang, Xiaoqi; Sogge, Christopher D (March 2025, Inventiones mathematicae)

   Abstract We obtain new optimal estimates for the$$L^{2}(M)\to L^{q}(M)$$ $L^2\left(M\right)\to L^q\left(M\right)$,$$q\in (2,q_{c}]$$ $q\in (2,q_c]$,$$q_{c}=2(n+1)/(n-1)$$ $q_c=2(n+1)/(n-1)$, operator norms of spectral projection operators associated with spectral windows$$[\lambda ,\lambda +\delta (\lambda )]$$ $[\lambda ,\lambda +\delta (\lambda \left)\right]$, with$$\delta (\lambda )=O((\log \lambda )^{-1})$$ $\delta \left(\lambda \right)=O\left({(log\lambda )}^{-1}\right)$on compact Riemannian manifolds$$(M,g)$$ $(M,g)$of dimension$$n\ge 2$$ $n\ge 2$all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of$$L^{q}$$ $L^q$-norms of quasimodes for each Lebesgue exponent$$q\in (2,q_{c}]$$ $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any$$q>q_{c}$$ $q>q_c$. 

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