Related papers: Regularity of higher codimension area minimizing i…
This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center…
In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the…
We give a new, simpler proof of the main approximation theorem for area minimizing current contained in Almgren's Big regularity paper. Our proof relies on a new estimate concerning the higher integrability of the quantity called here the…
This a survey on a series of recent papers in collaboration with Emanuele Spadaro on the regularity of area-minimizing currents in codimension higher than $1$.
This short note is the announcement of a forthcoming work in which we prove a first general boundary regularity result for area-minimizing currents in higher codimension, without any geometric assumption on the boundary, except that it is…
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,\alpha}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the…
We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the…
We establish a first general partial regularity theorem for area minimizing currents $\mathrm{mod}(p)$, for every $p$, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of…
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,\alpha}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the…
The interior regularity of area-minimizing integral currents and semi-calibrated currents has been studied extensively in recent decades, with sharp dimension estimates established on their interior singular sets in any dimension and…
Following Almgren's construction of the "center manifold" in his Big regularity paper, we show the C^{3,\alpha} regularity of area-minimizing currents in the neighborhood of points of density one without using the nonparametric theory. This…
We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and…
This paper is partly based on a lecture delivered by the author at the ERC workshop "Geometric Partial Differential Equations" held in Pisa in September 2012. What is presented here is an expanded version of that lecture.
In recent years considerable advances have been made in quantitative homogenization of partial differential equations in the periodic and non-periodic settings. This monograph surveys the theory of quantitative homogenization for…
De Lellis and coauthors have proved a sharp regularity theorem for area-minimizing currents in finite coefficient homology. They prove that area-minimizing mod $v$ currents are smooth outside of a singular set of codimension at least $1.$…
These are short notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVIIth Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in…
In the present work, we consider area minimizing currents in the general setting of arbitrary codimension and arbitrary boundary multiplicity. We study the boundary regularity of 2d area minimizing currents, beyond that, several results are…
We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,\alpha}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only…
We prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current;…
In analogy with Almgren's Theorem for area minimizing currents of general dimension and codimension, we prove that an $m$-dimensional semicalibrated current in a $(n+m)$-dimensional $C^{3,\varepsilon_0}$ manifold, semicalibrated by a…