Calculus of Variations and Geometric Measure Theory

G. De Philippis - E. Paolini

A short proof of the minimality of Simons cone

created by paolini on 26 Mar 2008
modified on 30 Nov 2016


Published Paper

Inserted: 26 mar 2008
Last Updated: 30 nov 2016

Journal: Rend. Sem. Mat. Univ. Padova
Volume: 121
Pages: 233-241
Year: 2009
Doi: 10.4171/RSMUP/121-14


In 1969 Bombieri, De Giorgi and Giusti proved that Simons cone is a minimal surface, thus providing the first example of a minimal surface with a singularity. We suggest a simplified proof of the same result. Our proof is based on the use of sub-calibrations, which are unit vector fields extending the normal vector to the surface, and having non-positive divergence. With respect to calibrations (which are divergence free) sub-calibrations are more easy to find and anyway are enough to prove the minimality of the surface.

Keywords: sub-calibrations, minimal surface, singularity, singular cone