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

Let $M = \sum_1 \times \sum_2 $ be the product of two compact Riemannian manifolds of dimension $n \ge 2$ and two, respectively. Let $\sum $ be the graph of a smooth map $f: \sum_1 \to \sum_2$, then $\sum$ is an n-dimensional submanifold of $M$. Let $\mathcal{G}$ be the Grassmannian bundle over $M$ whose fiber at each point is the set of all n-dimensional subspaces of the tangent space of $M$. The Gauss map $\gamma :\sum \to \mathcal{G}$ assigns to each point $x \in \sum $ the tangent space of \sum$ at x. This article considers the mean curvature flow of $\sum $ in $M$. When $\sum_1|$ and $\sum_2|$ are of the same non-negative curvature, we show a sub-bundle \mathcal{S}$ of the Grassmannian bundle is preserved along the flow, i.e. if the Gauss map of the initial submanifold $\sum $ lies in \mathcal{S}$, then the Gauss map of $\sum_t$ at any later time $t$ remains in $\mathcal{S}$. We also show that under this initial condition, the mean curvature flow remains a graph, exists for all time and converges to the graph of a constant map at infinity . As an application, we show that if $f$ is any map from $S^n$ to $S^2$ and if at each point, the restriction of $df$ to any two dimensional subspace is area decreasing, then f is homotopic to a constant map.

Let X be a compact connected strongly pseudoconvex CR Manifold of real dimension 2n.1 in Cn+1. Tanaka introduced a spec- tral sequence E(p,q) r (X) with E(p,q) 1 (X) being the Kohn-Rossi cohomology group and E(k,0) 2 (X) being the holomorphic De Rham cohomology denoted by Hk h(X). We study the holomorphic De Rham cohomology in terms of the s-invariant of the isolated sin- gularities of the variety V bounded by X. We give a characterization of the singularities with vanishing s-invariants. For n ≥ 3, Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n = 2, the problem has re- mained unsolved for over a quarter of a century. In this paper, we make progress in this direction by putting some conditions on X so that V will have very mild singularities. Specifically, we prove that if dimX = 3 and H2 h(X) = 0, then X is a boundary of complex variety V with only isolated quasi-homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational.

We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.