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

Yng-Ing Lee·Mu-Tao Wang
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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 $Σ$ 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.

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Yng-Ing Lee, Mu-Tao Wang. 2007-02-11. A Note on the Stability and Uniqueness for Solutions to the Minimal Surface System. https://arxiv.org/abs/math/0702303

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