Regularity of area minimizing currents. III: Blow-up
type: Item
instance of: scholarly article
title: Regularity of area minimizing currents. III: Blow-up
zbMATH Open document ID: 1345.49053
author: Camillo De Lellis
author: Emanuele Spadaro
published in: Annals of Mathematics. Second Series
publication date: 2016-04-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.1194
reviewed by: Vasile Oproiu
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49N60
zbMATH DE Number: 6575344
zbMATH Keywords: area minimization
zbMATH Keywords: integer rectifiable currents
zbMATH Keywords: regularity
MaRDI profile type: Publication
OpenAlex ID: W1822946441
arXiv ID: 1306.1194
cites work: Q4515158
cites work: Ordinary differential equations. An introduction to nonlinear analysis. Transl. from the German by Gerhard Metzen
cites work: 𝑄-valued functions revisited
cites work: Regularity of area minimizing currents I: gradient \(L^p\) estimates
cites work: Multiple valued functions and integral currents
cites work: Regularity of area minimizing currents. II: Center manifold
cites work: Q5565773
cites work: Riemannian geometry
cites work: Q5829984
cites work: Q3336268
cites work: Stratification of minimal surfaces, mean curvature flows, and harmonic maps.
DOI: 10.4007/ANNALS.2016.183.2.3
review text: This is third and last of a series of papers in which the authors give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds.NEWLINENEWLINEAssumption 0.1. Let \(\varepsilon_0 \in [0,1], m,\overline{n}\in \mathbb{N}\backslash \{0\}\) and \(l\in \mathbb{N}\). Denote by \((M)\) \(\Sigma \subset \mathbb{R}^{m+n} =\mathbb{R}^{m+\overline{n}+l}\) an embedded \((m+\overline{n})\)-dimensional submanifold of class \(C^{3,\varepsilon_0}\) and by \(T\) an integral current of dimension \(m\) with compact support \(\mathrm{spt}(T)\subset \Sigma\), which is area minimizing in \(\Sigma\).NEWLINENEWLINEFor a given \(T\) and \(\Sigma\) as in Assumption 0.1, define \(\mathrm{Reg}(T)=\{x\in \mathrm{spt}(T): \mathrm{spt}(T)\cap \mathbb{B}_r(x)\) is a \(C^{3,\varepsilon_0}\) submanifold for some \(r>0\)\}.NEWLINENEWLINENext, define \(\mathrm{Sing}(T) = \mathrm{spt}(T)\setminus (\mathrm{spt}(\partial T)\cup \mathrm{Reg} (T))\).NEWLINENEWLINEThe authors obtain an estimate on the Hausdorff dimension \(\mathrm{dim}_H(\mathrm{Sing}(T))\) of \(\mathrm{Sing}(T)\) by proving:NEWLINENEWLINETheorem 0.3. \(\mathrm{dim}_H(\mathrm{Sing}(T))\leq m-2\) for any \(m,\overline{n}, l, T\) and \(\Sigma \) as in Assumption 0.1.NEWLINENEWLINEThe authors start from the following Assumption 0.4. (Contradiction). There exist \(m\geq 2, \overline{n}, l, \Sigma\) and \(T\) as in Assumption 0.1 such that \(\mathcal{H}^{m-2+\alpha}(\mathrm{Sing}(T))>0\) for some \(\alpha>0\). Then they make a careful blow-up analysis split up into the following steps.NEWLINENEWLINE0.1. \textit{Flat tangent planes}. They reduce the flat blow-ups around a given point (assume that this point is the origin). The blow-ups will be chosen so that the size of the singular set satisfies an appropriate uniform estimate.NEWLINENEWLINE0.2. \textit{Intervals of flattening}. For appropriate rescalings of the current around the origin, one can use the center manifol…
description: scientific article; zbMATH DE number 6575344