First variation formula for generalized Gauss graphs
type: Item
instance of: scholarly article
title: First variation formula for generalized Gauss graphs
zbMATH Open document ID: 1032.49044
DOI: 10.1007/BF02505946
author: Silvano Delladio
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2003-10-19T00:00:00Z
review text: Using direct methods in the calculus of variations, \textit{G. Anzellotti, R. Serapioni}, and \textit{I. Tamanini} [Indiana Univ. Math. J. 39, 617-669 (1990; Zbl 0718.49030)] considered the problem of minimizing functionals \(\mathcal F (M)\), defined on a surface \(M\) and depending on the curvatures of \(M\). The key idea was to consider the graph \(G\) of the Gauss map of a surface \(M\), and to consider functionals \(\mathcal F\) such that Area \((G)\leq c \mathcal F(M)\). In [Boll. Unione Mat. Ital., VII. Ser., B 10, 991-1017 (1996; Zbl 0886.49031)] the present author introduced the notion of a generalized Gauss graph for any codimension in terms of integer multiplicity rectifiable currents, in particular, the Gauss graphs of smooth oriented submanifolds. In this paper, the author proves the divergence theorem, derives the first variation formula for generalized Gauss graphs, and studies some relations between mean curvature measure and first variation of the nonoriented varifold associated to a generalized Gauss graph.
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 1993714
zbMATH Keywords: Gauss graph
zbMATH Keywords: rectifiable currents
zbMATH Keywords: oriented curvature varifold
zbMATH Keywords: divergence theorem
zbMATH Keywords: first variation formula
reviewed by: Andrew Bucki
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1993714