It was in the early nineteen thirties that Douglas and Radd solved the Plateau problem for a rectifiable Jordan curve in Euclidean space. The minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk. In 1948 Morrey generalized the theorem of Douglas and Radd to minimal surfaces in a homogeneously regular Riemannian manifold. In both cases the solutions are defective as they may possess branch points. It was not until 1968 that Osserman [-24] was able to prove that interior branch points do not exist on the Douglas-Radd-Morrey solution.(Osserman's theorem in this full generality was proved by Alt [-2, 3] and Gulliver [8].) If the Jordan curve is real analytic, Gulliver and Lesley [9] showed that the solution surface is free of branch points even on its boundary. Hence, at least in this case, we know that the Douglas-Radd-Morrey solution is an immersion and represents a

Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants-k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justi-fication of such a statement.

Given a closed strictly convex hypersurface M in the Euclidean space I?"+*, the Gauss map of M defines a homeomorphism between M and the unit sphere S". Therefore the Gauss-Kronecker curvature of M can be transplanted via the Gauss map to a function defined on S". If this function is denoted by K, then Minkowski observed that K must satisfy the equation where xi are the coordinate functions on S". Minkowski then asked the converse of the problem. Namely, given a positive function K defined on S" satisfying the above integral conditions, can we find a closed strictly convex hypersurface whose curvature function is given by K? Minkowski solved the problem in the category of polyhedrons. Then AD Alexandrov and others solved the problem in general. However, this last solution does not provide any information about the regularity of the (unique) convex hypersurface even if we assume K is smooth. In the two

We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential equations constructed form the local geometry near a Fano surface within a Calabi-Yau manifold. We interpret the Gromov-Witten-type numbers from an enumerative point of view. We also describe the geometry of singular surfaces and show how the local invariants of singular surfaces agree with the smooth cases when they occur as complete intersections.

Hence the energy defines a functional on the space of Lipshitz maps between M and M'. Critical points of this functional are called harmonic maps. These maps were studied by Bochner, Morrey, Rauch, Eells and Sampson, Hartman, Uhlenbeck, Hamilton, Hildebrandt and others. The first fundamental result was due to Eells and Sampson [3] who proved that, in case M'has non-positive sectional curvature, each map from M to M'is homotopic to a harmonic map.(This result was then extended by Hamilton [8] to the case where both M and M'are allowed to have boundary.) Later Hartman [7] was able to prove the harmonic map is unique in each homotopy class if M'has strictly negative curvature. This last result of Hartman leads one to believe that harmonic maps between compact manifolds with negative curvature must enjoy a lot of nice properties. In fact, a few years ago, B. Lawson and the second author conjectured