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December  2011, 31(4): 1249-1272. doi: 10.3934/dcds.2011.31.1249

Center manifold: A case study

Camillo De Lellis 1, and  Emanuele Spadaro 2,

1. 

Universität Zürich, Institut für Mathematik, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland
2. 

Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Received  February 2010 Revised  January 2011 Published  September 2011

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 study is intended as a first step towards the understanding of Almgren's construction in its full generality.
Keywords: Area minimizing currents, regularity theory., minimal surfaces.
Mathematics Subject Classification: Primary: 49Q15, 49Q05; Secondary: 35D1.
Citation: Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
References:
[1]

Frederick J. Almgren, Jr., "Almgren's Big Regularity Paper. $Q$-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2," World Scientific Monograph Series in Mathematics, 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.  Google Scholar

[2]

Ennio De Giorgi, "Frontiere Orientate di Misura Minima," Seminario di Matematica della Scuola Normale Superiore di Pisa 1960-61, Editrice Tecnico Scientifica, Pisa, (1961), 57 pp.  Google Scholar

[3]

Camillo De Lellis and Emanuele Nunzio Spadaro, Higher integrability and approximation of minimal currents,, preprint, ().   Google Scholar

[4]

Lawrence C. Evans and Ronald F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar

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Eberhard Hopf, Über den funktionalen, insbesondere den analytischen Charakter der Lösun-gen elliptischer Differentialgleichungen zweiter Ordnung, Math. Z., 34 (1932), 194-233. doi: 10.1007/BF01180586.  Google Scholar

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Leon Simon, "Lectures on Geometric Measure Theory," Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983.  Google Scholar

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Elias M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

show all references

References:
[1]

Frederick J. Almgren, Jr., "Almgren's Big Regularity Paper. $Q$-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2," World Scientific Monograph Series in Mathematics, 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.  Google Scholar

[2]

Ennio De Giorgi, "Frontiere Orientate di Misura Minima," Seminario di Matematica della Scuola Normale Superiore di Pisa 1960-61, Editrice Tecnico Scientifica, Pisa, (1961), 57 pp.  Google Scholar

[3]

Camillo De Lellis and Emanuele Nunzio Spadaro, Higher integrability and approximation of minimal currents,, preprint, ().   Google Scholar

[4]

Lawrence C. Evans and Ronald F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar

[5]

Eberhard Hopf, Über den funktionalen, insbesondere den analytischen Charakter der Lösun-gen elliptischer Differentialgleichungen zweiter Ordnung, Math. Z., 34 (1932), 194-233. doi: 10.1007/BF01180586.  Google Scholar

[6]

Leon Simon, "Lectures on Geometric Measure Theory," Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983.  Google Scholar

[7]

Elias M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

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