# 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 - A, 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., (2000).   Google Scholar [2] Ennio De Giorgi, "Frontiere Orientate di Misura Minima,", Seminario di Matematica della Scuola Normale Superiore di Pisa 1960-61, (1961), 1960.   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, (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.  doi: 10.1007/BF01180586.  Google Scholar [6] Leon Simon, "Lectures on Geometric Measure Theory,", Proceedings of the Centre for Mathematical Analysis, 3 (1983).   Google Scholar [7] Elias M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43,, Monographs in Harmonic Analysis, (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., (2000).   Google Scholar [2] Ennio De Giorgi, "Frontiere Orientate di Misura Minima,", Seminario di Matematica della Scuola Normale Superiore di Pisa 1960-61, (1961), 1960.   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, (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.  doi: 10.1007/BF01180586.  Google Scholar [6] Leon Simon, "Lectures on Geometric Measure Theory,", Proceedings of the Centre for Mathematical Analysis, 3 (1983).   Google Scholar [7] Elias M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43,, Monographs in Harmonic Analysis, (1993).   Google Scholar
 [1] Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218 [2] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451 [3] Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460 [4] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [5] Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051 [6] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [7] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267 [8] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [9] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

2019 Impact Factor: 1.338