2019/03/31 by Mario Santilli · 1 citation
Mathematics · #Bounded function #Bounded variation #Contraction (grammar) #Curvature #Geometric Analysis and Curvature Flows #Mean curvature #Nonlinear Partial Differential Equations #Normal bundle #Point processes and geometric inequalities #Property (philosophy) #Uniform boundedness #math.AP #math.DG #msc:35D40 #msc:49Q10 #msc:49Q20 #msc:53A07 #msc:53C24
paper · pdf · doi:10.1142/s1664360720500083
published as Bulletin of Mathematical Sciences, 10(1), 2020 · One of the main results has been slightly improved (see 3.9) and a new corollary has been added (see 4.5). New title and abstract. The introduction has been also revised. To appear in Bull. Math. Sci
openalex created_date 2019/04/01 · openalex publication_date 2020/01/31 · arxiv created 2020/03/02 · arxiv updated 2020/07/20 · openalex updated_date 2026/08/05
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.