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The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces

https://doi.org/10.1002/cpa.22189

De_Philippis, Guido; De_Rosa, Antonio (July 2024, Communications on Pure and Applied Mathematics)

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min‐max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3.
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(Quasi-)conformal methods in two-dimensional free boundary problems

https://doi.org/10.4171/JEMS/1435

De_Philippis, Guido; Spolaor, Luca; Velichkov, Bozhidar (March 2024, Journal of the European Mathematical Society)

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On the Boundary Behavior of Mass-Minimizing Integral Currents

https://doi.org/10.1090/memo/1446

De_Lellis, Camillo; De_Philippis, Guido; Hirsch, Jonas; Massaccesi, Annalisa (November 2023, Memoirs of the American Mathematical Society)

Let $Σ<#comment/>$ be a smooth Riemannian manifold, $Γ<#comment/> \subset<#comment/> Σ<#comment/>$ a smooth closed oriented submanifold of codimension higher than $2$ and $T$ an integral area-minimizing current in $Σ<#comment/>$ which bounds $Γ<#comment/>$ . We prove that the set of regular points of $T$ at the boundary is dense in $Γ<#comment/>$ . Prior to our theorem the existence of any regular point was not known, except for some special choice of $Σ<#comment/>$ and $Γ<#comment/>$ . As a corollary of our theorem we answer to a question in Almgren’sAlmgren’s big regularity paperfrom 2000 showing that, if $Γ<#comment/>$ is connected, then $T$ has at least one point $p$ of multiplicity $\frac{1}{2}$ , namely there is a neighborhood of the point $p$ where $T$ is a classical submanifold with boundary $Γ<#comment/>$ ; we generalize Almgren’s connectivity theorem showing that the support of $T$ is always connected if $Γ<#comment/>$ is connected; we conclude a structural result on $T$ when $Γ<#comment/>$ consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when $Σ<#comment/> = R^{m + 1}$ and $T$ is $m$ -dimensional.
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