The isoperimetric inequality for a minimal submanifold in Euclidean space

Brendle, Simon

doi:10.1090/jams/969

The isoperimetric inequality for a minimal submanifold in Euclidean space
HTML articles powered by AMS MathViewer

by Simon Brendle;

J. Amer. Math. Soc. 34 (2021), 595-603

DOI: https://doi.org/10.1090/jams/969

Published electronically: February 18, 2021

HTML | PDF | Request permission

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 $2$.

References

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 130623, 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 C 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