(Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currents

Engelstein, Max¹ ; Spolaor, Luca¹ ; Velichkov, Bozhidar²

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States
² Laboratoire Jean Kuntzmann, Université Grenoble Alpes, Grenoble, France

Geometry & topology, Tome 23 (2019) no. 1, pp. 513-540.

Résumé

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing any given trace in the radial direction along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (eg work of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (eg integrability). If the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new regularity result for almost area-minimizing currents at singular points where at least one blowup is a multiplicity-one cone with isolated singularity. This result is similar to the one for stationary varifolds of Leon Simon (1983), but independent from it since almost-minimizers do not satisfy any equation.

Détail

DOI : 10.2140/gt.2019.23.513

Classification : 53A10
Keywords: regularity of minimal surfaces, epiperimetric inequality, almost area-minimizing currents

Bibliographie

[1] D Adams, L Simon, Rates of asymptotic convergence near isolated singularities of geometric extrema, Indiana Univ. Math. J. 37 (1988) 225 | DOI

[2] W K Allard, F J Almgren Jr., On the radial behavior of minimal surfaces and the uniqueness of their tangent cones, Ann. of Math. 113 (1981) 215 | DOI

[3] T H Colding, W P Minicozzi Ii, Uniqueness of blowups and Łojasiewicz inequalities, Ann. of Math. 182 (2015) 221 | DOI

[4] M Colombo, L Spolaor, B Velichkov, Direct epiperimetric inequalities for the thin obstacle problem and applications, preprint (2017)

[5] M Colombo, L Spolaor, B Velichkov, A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal. 28 (2018) 1029 | DOI

[6] C De Lellis, E Spadaro, L Spolaor, Uniqueness of tangent cones for two-dimensional almost-minimizing currents, Comm. Pure Appl. Math. 70 (2017) 1402 | DOI

[7] M Engelstein, L Spolaor, B Velichkov, Uniqueness of the blow-up at isolated singularities for the Alt–Caffarelli functional, preprint (2018)

[8] S Łojasiewicz, Ensembles semi-analytiques, IHES (1965)

[9] T Nagura, On the Jacobi differential operators associated to minimal isometric immersions of symmetric spaces into spheres, III, Osaka J. Math. 19 (1982) 241

[10] E R Reifenberg, An epiperimetric inequality related to the analyticity of minimal surfaces, Ann. of Math. 80 (1964) 1 | DOI

[11] R Schoen, L Simon, A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals, Indiana Univ. Math. J. 31 (1982) 415 | DOI

[12] L Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. 118 (1983) 525 | DOI

[13] L Simon, Lectures on geometric measure theory, 3, Australian National University (1983)

[14] J E Taylor, Regularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3 in R3, Invent. Math. 22 (1973) 119 | DOI

[15] J E Taylor, The structure of singularities in solutions to ellipsoidal variational problems with constraints in R3, Ann. of Math. 103 (1976) 541 | DOI

[16] B White, Tangent cones to two-dimensional area-minimizing integral currents are unique, Duke Math. J. 50 (1983) 143 | DOI