The isoperimetric inequality for a minimal submanifold in Euclidean space
Author:
Simon Brendle
Journal:
J. Amer. Math. Soc. 34 (2021), 595-603
MSC (2020):
Primary 53A07, 53A10
DOI:
https://doi.org/10.1090/jams/969
Published electronically:
February 18, 2021
MathSciNet review:
4280868
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Abstract: We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most $2$. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most $2$.
- William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. MR 307015, DOI 10.2307/1970868
- F. Almgren, Optimal isoperimetric inequalities, Indiana Univ. Math. J. 35 (1986), no. 3, 451–547. MR 855173, DOI 10.1512/iumj.1986.35.35028
- Xavier Cabré, Elliptic PDE’s in probability and geometry: symmetry and regularity of solutions, Discrete Contin. Dyn. Syst. 20 (2008), no. 3, 425–457. MR 2373200, DOI 10.3934/dcds.2008.20.425
- Torsten Carleman, Zur Theorie der Minimalflächen, Math. Z. 9 (1921), no. 1-2, 154–160 (German). MR 1544458, DOI 10.1007/BF01378342
- Philippe Castillon, Submanifolds, isoperimetric inequalities and optimal transportation, J. Funct. Anal. 259 (2010), no. 1, 79–103. MR 2610380, DOI 10.1016/j.jfa.2010.03.001
- Jaigyoung Choe, The isoperimetric inequality for a minimal surface with radially connected boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 4, 583–593. MR 1093710
- Jaigyoung Choe, Isoperimetric inequalities of minimal submanifolds, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 325–369. MR 2167266
- Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
- Jerry M. Feinberg, The isoperimetric inequality for doubly-connected minimal surfaces in $\textbf {R}^{n}$, J. Analyse Math. 32 (1977), 249–278. MR 461306, DOI 10.1007/BF02803583
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
- Chuan-chih Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. (2) 73 (1961), 213–220. MR 130637, DOI 10.2307/1970287
- Peter Li, Richard Schoen, and Shing-Tung Yau, On the isoperimetric inequality for minimal surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 2, 237–244. MR 764944
- J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 344978, DOI 10.1002/cpa.3160260305
- R. Osserman and M. Schiffer, Doubly-connected minimal surfaces, Arch. Rational Mech. Anal. 58 (1975), no. 4, 285–307. MR 385687, DOI 10.1007/BF00250292
- William T. Reid, The isoperimetric inequality and associated boundary problems, J. Math. Mech. 8 (1959), 897–905. MR 0130623, DOI 10.1512/iumj.1959.8.58057
- Andrew Stone, On the isoperimetric inequality on a minimal surface, Calc. Var. Partial Differential Equations 17 (2003), no. 4, 369–391. MR 1993960, DOI 10.1007/s00526-002-0174-9
- Peter Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv. 83 (2008), no. 3, 539–546. MR 2410779, DOI 10.4171/CMH/135
- Neil S. Trudinger, Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 4, 411–425 (English, with English and French summaries). MR 1287239, DOI 10.1016/S0294-1449(16)30181-0
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Additional Information
Simon Brendle
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
MR Author ID:
655348
Received by editor(s):
July 26, 2019
Received by editor(s) in revised form:
May 19, 2020, and September 5, 2020
Published electronically:
February 18, 2021
Additional Notes:
This project was supported by the National Science Foundation under grant DMS-1806190 and by the Simons Foundation.
Article copyright:
© Copyright 2021
American Mathematical Society