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In 1976, the author (see the open problem section in [3]) asked the following question: if p is a function defined on i? 3, what is the condition on p so that there is a sphere in R3 whose mean curvature is given by p. There were important works by Treibergs-Wei [2] and others on this question. Recently in a lecture that I gave in the Chinese University of Hong Kong, I realized that my work with R. Schoen on the existence of black hole can be used to settle the prescribed mean curvature question for a large class of p.

We discuss a method for deriving differential equations for the prepotential and the mirror map arising from a pair of Calabi-Yau manifold. Examples with one Kahler modulus are given. Here we nd that the differential equations we derive almost entirely characterize the prepotential and the mirror map in question. Some of the results here have been announced previously in

Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on discrete exterior calculus methods using hybrid domains. We then analyze the geometric structure of curvature operators in a piecewise-flat lattice.

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