NRODUCTION LET y be a rectifiable Jordan curve in three-dimensional euclidean space. Answering an old question, whether y can bound a surface with minimal area, Douglas [l I] and Rad6 [45](independently) found a minimal surface spanning y which is parametrized by the disk. This minimal surface has minimal area among all Lipschitz maps from the disk into R3 which span y. The question whether this solution has branch points or not was finally settled by Osserman [42], who proved that there are no interior true branch points, and by Gulliver [lS], who proved that there are no interior false branch points.

We shall consider the following Dirichlet eigenvalue problem on a smooth bounded domain S~ eRn, I where V is a nonnegative function defined on,~. As is well-known, the eigenvalues of problem (1.1) can be interpreted as the energy levels of a particle travelling under an external force field of a potential q in Rn, where and the corresponding eigenfunctions are wave functions of the Schrodinger equation-J~+ qu= lu. Furthermore, the set of eigenvalues {A,} of (1.1) are nonnegative and can be arranged in a nondecreasing order as follows, It is a significant problem to find a lower bound for~, 1 in terms of the geometry of Q. This subject has been studied extensively by many authors. A rather precise bound in the case V== 0 was worked out not only for a bounded domain in but actually valid for a general Riemannian manifold with certain curvature conditions; we refer to [4] for these recent develop-ments. Nevertheless, very little is known about the obvious interesting question of how big the gap is between 2 and l. There are both physical

This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology bases from simplicial complex structures, then construct dual cohomology bases and diffuse them to harmonic 1-forms. Next, we construct bases of holomorphic differentials. We then obtain period matrices by integrating holomorphic differentials along homology bases. We also study the global conformal mapping between genus zero surfaces and spheres, and between general meshes and planes. Our method of computing conformal structures can be applied to tackle fundamental problems in computer aid design and computer graphics, such as geometry classification and identification, and surface global parametrization.

The purpose of this paper is to solve a problem posed by Strominger in constructing smooth models of superstring theory with flux. These are given by non-Khler manifolds with torsion.

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