Mathematical Research Letters

Volume 15, Issue 1, January 2008 pp. 197-206.

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

Authors: Yng-Ing Lee (1) and Mu-Tao Wang (2)
Author institution: National Taiwan University (1) and Columbia University (2)

Summary: 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. The complete statements of the results appear in Theorem~3.1, Theorem~3.2, and Theorem~4.1.

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