Regularity of area minimizing currents mod p

We establish a first general partial regularity theorem for area minimizing currents mod ( p ) , for every p , in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m -dimensional area minimizing current mod ( p ) cannot be larger...

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Published in	Geometric and functional analysis Vol. 30; no. 5; pp. 1224 - 1336
Main Authors	De Lellis, Camillo, Hirsch, Jonas, Marchese, Andrea, Stuvard, Salvatore
Format	Journal Article
Language	English
Published	Cham Springer International Publishing 01.10.2020
Subjects	Analysis Mathematics Mathematics and Statistics Area minimizing currents Minimal surfaces Center manifold 49N60 49Q05 49Q15 Regularity theory 35B65 Multiple valued functions Blow-up analysis 35J47
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Abstract	We establish a first general partial regularity theorem for area minimizing currents mod ( p ) , for every p , in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m -dimensional area minimizing current mod ( p ) cannot be larger than m - 1 . Additionally, we show that, when p is odd, the interior singular set is ( m - 1 ) -rectifiable with locally finite ( m - 1 ) -dimensional measure.
AbstractList	We establish a first general partial regularity theorem for area minimizing currents mod ( p ) , for every p , in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m -dimensional area minimizing current mod ( p ) cannot be larger than m - 1 . Additionally, we show that, when p is odd, the interior singular set is ( m - 1 ) -rectifiable with locally finite ( m - 1 ) -dimensional measure. We establish a first general partial regularity theorem for area minimizing currents $${\mathrm{mod}}(p)$$ mod ( p ) , for every p , in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m -dimensional area minimizing current $${\mathrm{mod}}(p)$$ mod ( p ) cannot be larger than $$m-1$$ m - 1 . Additionally, we show that, when p is odd, the interior singular set is $$(m-1)$$ ( m - 1 ) -rectifiable with locally finite $$(m-1)$$ ( m - 1 ) -dimensional measure.
Author	Stuvard, Salvatore Hirsch, Jonas Marchese, Andrea De Lellis, Camillo
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References	De LellisCamilloSpadaroEmanueleRegularity of area minimizing currents III: blow-upAnn. of Math. (2)20161832577617345048310.4007/annals.2016.183.2.3 De LellisCamilloSpadaroEmanuele NunzioRegularity of area minimizing currents II: center manifoldAnn. of Math. (2)20161832499575345048210.4007/annals.2016.183.2.2 SimonLCylindrical tangent cones and the singular set of minimal submanifoldsJournal of Differential Geometry1993383585652124378810.4310/jdg/1214454484 TaylorJean ERegularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document} in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^{3}$$\end{document}Invent. Math.19732211915933390310.1007/BF01392299 WhiteBrianStratification of minimal surfaces, mean curvature flows, and harmonic mapsJ. Reine Angew. Math.199748813514653650874.58007 AmbrosioLuigiFuscoNicolaPallaraDiegoFunctions of bounded variation and free discontinuity problems2000Oxford University Press, New YorkOxford Mathematical Monographs. The Clarendon Press0957.49001 HerbertFederer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. FrederickJ. Almgren, Jr. Multi-functions mod ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}. In Geometric analysis and computer graphics (Berkeley, CA, 1988), volume 17 of Math. Sci. Res. Inst. Publ., pages 1–17. Springer, New York, 1991. SpolaorLucaAlmgren’s type regularity for semicalibrated currentsAdv. Math.2019350747815394868510.1016/j.aim.2019.04.057 KrantzSteven GParksHarold RGeometric integration theory2008Boston, MACornerstones. Birkhäuser Boston Inc10.1007/978-0-8176-4679-0 BrianWhite. A regularity theorem for minimizing hypersurfaces modulo p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}. In Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 413–427. Amer. Math. Soc., Providence, RI, 1986. MarcheseAndreaStuvardSalvatoreOn the structure of flat chains modulo p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}Adv. Calc. Var.2018113309323381952910.1515/acv-2016-0040 De PauwThierryHardtRobertRectifiable and flat G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} chains in a metric spaceAmer. J. Math.20121341169287613810.1353/ajm.2012.0004 YoungRobertQuantitative nonorientability of embedded cyclesDuke Math. J.2018167141108374369910.1215/00127094-2017-0035 De PauwThierryHardtRobertSome basic theorems on flat G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} chainsJ. Math. Anal. Appl.2014418210471061320669710.1016/j.jmaa.2014.04.024 CamilloDe Lellis, JonasHirsch, AndreaMarchese, and SalvatoreStuvard. Area minimizing currents mod 2Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2Q}$$\end{document}: linear regularity theory. Comm. Pure Appl. Math., to appear. arXiv:1909.03305. De LellisCamilloSpadaroEmanuele NunzioQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document}-valued functions revisitedMem. Amer. Math. Soc.2011211991vi+7926637351246.49001 Some open problems in geometric measure theory and its applications suggested by participants of the 1984 AMS summer institute. In J. E. Brothers, editor, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 441–464. Amer. Math. Soc., Providence, RI, 1986. PerttiMattila. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. RichardSchoen, LeonSimon, and FrederickJ. Jr. Almgren. Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II. Acta Math., 139(3-4):217–265, 1977. De LellisCamilloSpadaroEmanueleRegularity of area minimizing currents I: gradient Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} estimatesGeom. Funct. Anal.201424618311884328392910.1007/s00039-014-0306-3 AmbrosioLuigiMetric space valued functions of bounded variationAnn. Scuola Norm. Sup. Pisa Cl. Sci. (4)199017343947910799850724.49027 AaronNaber and DanieleValtorta. The singular structure and regularity of stationary varifolds. J. Eur. Math. Soc. (JEMS), to appear. arXiv:1505.03428. LeonSimon. Rectifiability of the singular sets of multiplicity 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1$$\end{document} minimal surfaces and energy minimizing maps. In Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), pages 246–305. Int. Press, Cambridge, MA, 1995. De LellisCamilloSpadaroEmanueleMultiple valued functions and integral currentsAnn. Sc. Norm. Super. Pisa Cl. Sci. (5)20151441239126934676551343.49073 FedererHerbertThe singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimensionBull. Amer. Math. Soc.19707676777126098110.1090/S0002-9904-1970-12542-3 AmbrosioLuigiKirchheimBerndCurrents in metric spacesActa Math.20001851180179418510.1007/BF02392711 LeonSimon. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983. AllardWilliam KOn the first variation of a varifoldAnn. of Math.197229541749130701510.2307/1970868 WhiteBrianThe structure of minimizing hypersurfaces mod 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4$$\end{document}Invent. Math.1979531455853868310.1007/BF01403190 Brian White (546_CR27) 1979; 53 Brian White (546_CR29) 1997; 488 Herbert Federer (546_CR16) 1970; 76 Robert Young (546_CR30) 2018; 167 Thierry De Pauw (546_CR14) 2014; 418 546_CR1 546_CR21 546_CR3 546_CR20 Luigi Ambrosio (546_CR6) 2000; 185 546_CR23 Luigi Ambrosio (546_CR5) 2000 546_CR24 L Simon (546_CR22) 1993; 38 Luigi Ambrosio (546_CR4) 1990; 17 546_CR28 Camillo De Lellis (546_CR8) 2014; 24 Camillo De Lellis (546_CR12) 2011; 211 Andrea Marchese (546_CR18) 2018; 11 546_CR7 Camillo De Lellis (546_CR11) 2016; 183 Camillo De Lellis (546_CR10) 2016; 183 Steven G Krantz (546_CR17) 2008 Luca Spolaor (546_CR25) 2019; 350 William K Allard (546_CR2) 1972; 2 Camillo De Lellis (546_CR9) 2015; 14 Thierry De Pauw (546_CR13) 2012; 134 546_CR15 546_CR19 Jean E Taylor (546_CR26) 1973; 22
References_xml	– reference: LeonSimon. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983. – reference: AmbrosioLuigiMetric space valued functions of bounded variationAnn. Scuola Norm. Sup. Pisa Cl. Sci. (4)199017343947910799850724.49027 – reference: De LellisCamilloSpadaroEmanuele NunzioRegularity of area minimizing currents II: center manifoldAnn. of Math. (2)20161832499575345048210.4007/annals.2016.183.2.2 – reference: YoungRobertQuantitative nonorientability of embedded cyclesDuke Math. J.2018167141108374369910.1215/00127094-2017-0035 – reference: De LellisCamilloSpadaroEmanueleRegularity of area minimizing currents I: gradient Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} estimatesGeom. Funct. Anal.201424618311884328392910.1007/s00039-014-0306-3 – reference: AmbrosioLuigiFuscoNicolaPallaraDiegoFunctions of bounded variation and free discontinuity problems2000Oxford University Press, New YorkOxford Mathematical Monographs. The Clarendon Press0957.49001 – reference: BrianWhite. A regularity theorem for minimizing hypersurfaces modulo p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}. In Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 413–427. Amer. Math. Soc., Providence, RI, 1986. – reference: AllardWilliam KOn the first variation of a varifoldAnn. of Math.197229541749130701510.2307/1970868 – reference: AaronNaber and DanieleValtorta. The singular structure and regularity of stationary varifolds. J. Eur. Math. Soc. (JEMS), to appear. arXiv:1505.03428. – reference: De PauwThierryHardtRobertSome basic theorems on flat G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} chainsJ. Math. Anal. Appl.2014418210471061320669710.1016/j.jmaa.2014.04.024 – reference: LeonSimon. Rectifiability of the singular sets of multiplicity 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1$$\end{document} minimal surfaces and energy minimizing maps. In Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), pages 246–305. Int. Press, Cambridge, MA, 1995. – reference: HerbertFederer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. – reference: MarcheseAndreaStuvardSalvatoreOn the structure of flat chains modulo p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}Adv. Calc. Var.2018113309323381952910.1515/acv-2016-0040 – reference: PerttiMattila. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. – reference: CamilloDe Lellis, JonasHirsch, AndreaMarchese, and SalvatoreStuvard. Area minimizing currents mod 2Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2Q}$$\end{document}: linear regularity theory. Comm. Pure Appl. Math., to appear. arXiv:1909.03305. – reference: Some open problems in geometric measure theory and its applications suggested by participants of the 1984 AMS summer institute. In J. E. Brothers, editor, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 441–464. Amer. Math. Soc., Providence, RI, 1986. – reference: De LellisCamilloSpadaroEmanueleMultiple valued functions and integral currentsAnn. Sc. Norm. Super. Pisa Cl. Sci. (5)20151441239126934676551343.49073 – reference: SpolaorLucaAlmgren’s type regularity for semicalibrated currentsAdv. Math.2019350747815394868510.1016/j.aim.2019.04.057 – reference: RichardSchoen, LeonSimon, and FrederickJ. Jr. Almgren. Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II. Acta Math., 139(3-4):217–265, 1977. – reference: WhiteBrianStratification of minimal surfaces, mean curvature flows, and harmonic mapsJ. Reine Angew. Math.199748813514653650874.58007 – reference: KrantzSteven GParksHarold RGeometric integration theory2008Boston, MACornerstones. Birkhäuser Boston Inc10.1007/978-0-8176-4679-0 – reference: De LellisCamilloSpadaroEmanueleRegularity of area minimizing currents III: blow-upAnn. of Math. (2)20161832577617345048310.4007/annals.2016.183.2.3 – reference: FrederickJ. Almgren, Jr. Multi-functions mod ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}. In Geometric analysis and computer graphics (Berkeley, CA, 1988), volume 17 of Math. Sci. Res. Inst. Publ., pages 1–17. Springer, New York, 1991. – reference: AmbrosioLuigiKirchheimBerndCurrents in metric spacesActa Math.20001851180179418510.1007/BF02392711 – reference: De LellisCamilloSpadaroEmanuele NunzioQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document}-valued functions revisitedMem. Amer. Math. Soc.2011211991vi+7926637351246.49001 – reference: De PauwThierryHardtRobertRectifiable and flat G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} chains in a metric spaceAmer. J. Math.20121341169287613810.1353/ajm.2012.0004 – reference: WhiteBrianThe structure of minimizing hypersurfaces mod 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4$$\end{document}Invent. Math.1979531455853868310.1007/BF01403190 – reference: FedererHerbertThe singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimensionBull. Amer. Math. Soc.19707676777126098110.1090/S0002-9904-1970-12542-3 – reference: SimonLCylindrical tangent cones and the singular set of minimal submanifoldsJournal of Differential Geometry1993383585652124378810.4310/jdg/1214454484 – reference: TaylorJean ERegularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document} in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^{3}$$\end{document}Invent. Math.19732211915933390310.1007/BF01392299 – volume: 22 start-page: 119 year: 1973 ident: 546_CR26 publication-title: Invent. 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Snippet	We establish a first general partial regularity theorem for area minimizing currents mod ( p ) , for every p , in any dimension and codimension. More... We establish a first general partial regularity theorem for area minimizing currents $${\mathrm{mod}}(p)$$ mod ( p ) , for every p , in any dimension and...
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Title	Regularity of area minimizing currents mod p
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