A Note on the Stability and Uniqueness for Solutions to the Minimal Surface System
Resource
arXiv: 0702303
Journal
Mathematical Research Letters
Pages
197-206
Date Issued
2007
Date
2007
Author(s)
Abstract
In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold is the graph of a (strictly) distance-decreasing map, then is (strictly) stable. It is known that a minimal graph of codimension one is stable without assuming the distance-decreasing condition. We give another criterion for the stability in terms of the two-Jacobians of the map which in particular covers the codimension one case. All theorems are proved in the more general setting for minimal maps between Riemannian manifolds. The complete statements of the results appear in Theorem 3.1, Theorem 3.2, and Theorem 4.1.
Type
journal article
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