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The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points

https://doi.org/10.15781/y8xg-xv26

De_Lellis, Camillo; Skorobogatova, Anna (January 2025, Ars inveniendi analytica)

Free, publicly-accessible full text available January 1, 2026
Excess decay for minimizing hypercurrents mod $2 Q$

https://doi.org/10.1016/j.na.2024.113606

De_Lellis, Camillo; Hirsch, Jonas; Marchese, Andrea; Spolaor, Luca; Stuvard, Salvatore (October 2024, Nonlinear Analysis)

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The regularity theory for the area functional (in geometric measure theory)

De_Lellis, Camillo (December 2023, European Mathematical Society Publishing House)
Beliaev, D; Smirnov, S (Ed.)
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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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