Let $ f : \sum_1 \to \sum_2$ be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in $\sum_1 \times \sum_2$2 by the mean curvature flow. Under suitable conditions on the curvature of $\sum_1$ and $\sum_2$ and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant $t$, the flow remains the graph of a map $f_t$ and $f_t$ converges to a constant map as $t$ approaches infinity. This also provides a regularity estimate for Lipschtz initial data.

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We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of [8] and [11].

The actions of a half Virasoro algebra have appeared in many integrable systems. In this paper we show that there is an action of a (Half) Virasoro algebra on the space of (2+0) harmonic maps into a Lie group. This action is generated by a natural action on the frames. A similar calculation on the space-time (1+1) harmonic maps yields formulas generated by John Schwarz.

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