\(\lambda\) minimizing currents
In geometric measure theory the notion of an integer multiplicity rectifiable current \(T\) of dimension \(n\) in \(\mathbb{R}^{n+k}\) is a generalization of the notion of an \(n\)-dimensional compact oriented submanifold with boundary. It has turned out to be very useful in the geometric calculus of variations, especially in the study of higher- dimensional minimal surfaces. Given an open subset \(U\) of \(\mathbb{R}^{n+k}\), and a nonnegative number \(\lambda\), the authors call such a current \(T\) \(\lambda\)-minimizing in \(U\), if the inequality \(M (T)\leq M(T+ \partial Q)+ \lambda M (Q)\) holds for every integer multiplicity rectifiable current \(Q\) of dimension \(n+1\) with support in \(U\). Here, \(M (\cdot)\) denotes the mass (the total area -- with multiplicities -- in the corresponding dimension), and \(\lambda=0\) is the intensively studied case of absolutely mass minimizing currents. \(\lambda\)-minimizing currents arise e.g. in the study of energy minimizing codimension one currents with prescribed mean curvature vector. In this paper, the authors present a systematic study of \(\lambda\)- minimizing currents. They prove monotonicity formulae, the fundamental inequalities between the cylindrical excess and the height over an \(n\)- dimensional plane in \(\mathbb{R}^{n+k}\). From these more technical results they are able to deduce graph representations of \(\lambda\)-minimizing currents with small excess as well as interior regularity theorems. These theorems apply to \(\lambda\)-minimizing currents of codimension one which are oriented frontiers in the sense of De Giorgi or which have Lipschitz mean curvature. An important tool in all the proofs is the famous regularity theorem of \textit{W. K. Allard} [Ann. Math., II. Ser. 95, 417- 491 (1972; Zbl 0252.49028)].
scientific article; zbMATH DE number 492794
scientific article