Calculus of Variations and Geometric Measure Theory
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M. Santilli

Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

created by santilli on 21 Jul 2020


Published Paper

Inserted: 21 jul 2020
Last Updated: 21 jul 2020

Journal: Bull. Math. Sci.
Volume: 10(1) 2020
Year: 2019
Doi: 10.1142/S1664360720500083


In this paper we deal with a class of varieties of bounded mean curvature in the viscosity sense that has the remarkable property to contain the blow up sets of all sequences of varifolds whose mean curvatures are uniformly bounded and whose boundaries are uniformly bounded on compact sets. We investigate the second-order properties of these varieties, obtaining results that are new also in the varifold’s setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, a property that allows to prove a Coarea-type formula for their generalized Gauss map. Then we use this formula to extend a sharp geometric inequality of Almgren and the associated soap bubble theorem. As a consequence of the geometric inequality we obtain sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled.

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