A Note on the Stability and Uniqueness for Solutions to the Minimal Surface System

Details A Note on the Stability and Uniqueness for Solutions to the Minimal Surface System A Note on the Stability and Uniqueness for Solutions to the Minimal Surface System

2007-02-11

arxiv.org/abs/math/0702303v1

Mathematics

Differential Geometry

13 pages

Scientific paper

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 $\Sigma$ is the graph of a (strictly) distance-decreasing map, then $\Sigma$ 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.

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