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1. Global Regularity of Skew Mean Curvature Flow for Small Data in d ≥ 4 Dimensions

   https://doi.org/10.1093/imrn/rnad104  

   Huang, Jiaxi; Li, Ze; Tataru, Daniel (May 2023, International Mathematics Research Notices)

   Abstract The skew mean curvature flow is an evolution equation for a $$d$$ dimensional manifold immersed into $$\mathbb {R}^{d+2}$$, and which moves along the binormal direction with a speed proportional to its mean curvature. In this article, we prove small data global regularity in low-regularity Sobolev spaces for the skew mean curvature flow in dimensions $$d\geq 4$$. This extends the local well-posedness result in [7]. 

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2. 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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3. 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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4. Minkowski Inequality in Cartan–Hadamard Manifolds

   https://doi.org/10.1093/imrn/rnad114  

   Ghomi, Mohammad; Spruck, Joel (June 2023, International Mathematics Research Notices)

   Using harmonic mean curvature flow, we establish a sharp Minkowski-type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard $$3$$-manifolds. This inequality also improves the known estimates for total mean curvature in hyperbolic $$3$$-space. As an application, we obtain a Bonnesen-style isoperimetric inequality for surfaces with convex distance function in nonpositively curved $$3$$-spaces, via monotonicity results for total mean curvature. This connection between the Minkowski and isoperimetric inequalities is extended to Cartan–Hadamard manifolds of any dimension. 

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5. Rigidity properties of Colding–Minicozzi entropies

   https://doi.org/10.1515/ans-2022-0082  

   Bernstein, Jacob (February 2024, Advanced Nonlinear Studies)

   Abstract We show certain rigidity for minimizers of generalized Colding–Minicozzi entropies. The proofs are elementary and work even in situations where the generalized entropies are not monotone along mean curvature flow. 

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