# American Institute of Mathematical Sciences

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

## Center manifold: A case study

 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.
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 [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

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
 [1] Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010 [2] Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54. [3] José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178 [4] Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123. [5] Francisco Brito, Maria Luiza Leite, Vicente de Souza Neto. Liouville's formula under the viewpoint of minimal surfaces. Communications on Pure & Applied Analysis, 2004, 3 (1) : 41-51. doi: 10.3934/cpaa.2004.3.41 [6] Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75 [7] Francesco Maggi, Salvatore Stuvard, Antonello Scardicchio. Soap films with gravity and almost-minimal surfaces. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6877-6912. doi: 10.3934/dcds.2019236 [8] Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079 [9] Giovanni Bellettini, Matteo Novaga, Giandomenico Orlandi. Eventual regularity for the parabolic minimal surface equation. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5711-5723. doi: 10.3934/dcds.2015.35.5711 [10] Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463 [11] Xifeng Su, Rafael de la Llave. On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7057-7080. doi: 10.3934/dcds.2019295 [12] Andrea Cianchi, Vladimir Maz'ya. Global gradient estimates in elliptic problems under minimal data and domain regularity. Communications on Pure & Applied Analysis, 2015, 14 (1) : 285-311. doi: 10.3934/cpaa.2015.14.285 [13] Gernot Greschonig. Regularity of topological cocycles of a class of non-isometric minimal homeomorphisms. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4305-4321. doi: 10.3934/dcds.2013.33.4305 [14] Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure & Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407 [15] Sun-Sig Byun, Hongbin Chen, Mijoung Kim, Lihe Wang. Lp regularity theory for linear elliptic systems. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 121-134. doi: 10.3934/dcds.2007.18.121 [16] Filippo Morabito. Singly periodic free boundary minimal surfaces in a solid cylinder of $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4987-5001. doi: 10.3934/dcds.2015.35.4987 [17] Mehmet Onur Fen, Marat Akhmet. Impulsive SICNNs with chaotic postsynaptic currents. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1119-1148. doi: 10.3934/dcdsb.2016.21.1119 [18] Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591 [19] Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 [20] Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425

2020 Impact Factor: 1.392

## Metrics

• PDF downloads (72)
• HTML views (0)
• Cited by (5)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]