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Mean curvature ows of hypersurfaces have been extensively studied and there are various dierent approaches and many beautiful results. However, relatively little is known about mean curvature ows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then introduces mean curvature ows in general dimensions and co-dimensions. The related techniques in the so called \blow-up" analysis are also discussed. At the end, we present some global existence and convergence results for mean curvature ows of two-dimensional surfaces in four-dimensional ambient spaces.

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ℂ^{$N$}. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter,$Math. Ann.$$337$(2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ℂ^{$N$}. Our results extend to certain weights and measures defined on cones in ℝ^{$N$}.

No abstract uploaded!

No abstract uploaded!