Almgren regularity theorem
In geometric measure theory, a field of mathematics, the Almgren regularity theorem, proved by Almgren (1983, 2000), states that the singular set of a mass-minimizing surface has codimension at least 2. Almgren's proof of this was 955 pages long. Within the proof many new ideas are introduced, such as monotonicity of a frequency function and the use of a center manifold to perform a more intricate blow-up procedure.
A streamlined and more accessible proof of Almgren's regularity theorem, following the same ideas as Almgren, was given by Camillo De Lellis and Emanuele Spadaro in a series of three papers.[1]
References
- De Lellis, Camillo; Spadaro, Emanuele Regularity of area minimizing currents III: blow-up. Ann. of Math. (2) 183 (2016), no. 2, 577–617.
- Almgren, F. J. (1983), "Q valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two", Bulletin of the American Mathematical Society, New Series, 8 (2): 327–328, doi:10.1090/S0273-0979-1983-15106-6, ISSN 0002-9904, MR 0684900
- Almgren, Frederick J. Jr. (2000), Taylor, Jean E.; Scheffer, Vladimir (eds.), 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, vol. 1, River Edge, NJ: World Scientific, ISBN 978-981-02-4108-7, MR 1777737, Zbl 0985.49001
- Chang, Sheldon X. (1998), "On Almgren's regularity result", The Journal of Geometric Analysis, 8 (5): 703–708, doi:10.1007/BF02922666, ISSN 1050-6926, MR 1731058, S2CID 120598029
- White, Brian (1998), "The mathematics of F. J. Almgren, Jr", The Journal of Geometric Analysis, 8 (5): 681–702, CiteSeerX 10.1.1.120.4639, doi:10.1007/BF02922665, ISSN 1050-6926, MR 1731057, S2CID 122083638
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