We prove the mean curvature flow of the graph of a symplectomorphism between Riemann surfaces converges smoothly as time approaches infinity.

The chapter discusses the group actions on R³. The Smith conjecture has many equivalent forms, and each of these forms has various consequences and generalizations. The Smith conjecture is a structure theorem about symmetries of the product of a compact surface with an interval. Here the interval may be closed or open. The usual Smith conjecture is equivalent to proving the smooth <i>Z_n</i> actions on <i>S</i>² [0, 1] are conjugate to actions that preserve the product structure. This generalized Smith conjecture represents the belief that all the symmetries of the product of a compact surface with an interval actually arise from the symmetries of the surface extended trivially to the product structure. The chapter presents how to apply minimal surfaces to study the generalized Smith conjecture on <i>S</i>² <i>I</i>, where <i>I</i> is an open or closed interval. A geodesic version of the loop theorem and a new equivariant

In this paper, we consider the heat ‡flow for p-pseudoharmonic maps from a closed Sasakian manifold into a compact Riemannian manifold. We prove global existence and asymptotic convergence of the solution for the p-pseudoharmonic map heat ‡flow provided that the sectional curvature of the target manifold is nonpositive. Moreover, without the curvature assumption on the target manifold, we obtain global existence and asymptotic convergence of the p-pseudoharmonic map heat fl‡ow as well when its initial p-energy is sufficiently small.

In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective benchmarks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.

In this paper we start the classification of strongly bracket-generated sub-symmetric spaces. We prove a structure theorem and analyze the nilpotent case.