2007/02/11 by Yng-Ing Lee, Mu-Tao Wang, Lee, Yng-Ing +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.math/0702303
13 pages
arxiv created 2007/02/11 · arxiv updated 2009/12/01
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 Σ is the graph of a (strictly) distance-decreasing map, then Σ 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.