Calculus of Variations and Geometric Measure Theory

G. De Philippis - F. Maggi

Sharp stability inequalities for the Plateau problem

created by maggi on 19 Dec 2011
modified by dephilipp on 02 Nov 2013


Accepted Paper

Inserted: 19 dec 2011
Last Updated: 2 nov 2013

Journal: J. Differential Geom.
Year: 2011


The validity of global quadratic stability inequalities for uniquely regular area minimizing hypersurfaces is proved to be equivalent to the uniform positivity of the second variation of the area. Concerning singular area minimizing hypersurfaces, by a “quantitative calibration” argument we prove quadratic stability inequalities with explicit constants for all the Lawson’s cones, excluding six exceptional cases. As a by- product of these results, explicit lower bounds for the first eigenvalues of the second variation of the area on these cones are derived.