Representation formulas of Weierstrass type in submanifold theory

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We investigated a global properties of Weierstrass representation. First of all, we studied a fundamental problem related to global problems. In particular, as the monodromy problem for minimal surfaces in Euclidean geometry is considered as a period problem of certain integral of holomorphic forms, that of CMC-1 surfaces in hyperbolic space can be considerd as a monodromy problem of a ordinary differential equation on Riemann surfaces. In this context, the monodromy problem is the condition for SL(2, C)-monodromy group to be reduced to the unitary group. To find a criterion of such a condition is difficult. However, when a problem satisfies some symmetric properties, it can be solved explicitly. Using this fact, we have constructed a large amount of examples of CMC-1 surfaces. Related to this problem, we investigated metrics of constant positive curvature with conical singularities, and obtained a classification result.Related to classification of CMC-1 surfaces, we defined a homology invariant on CMC-1 surface, which is called flux, and proved some non-existence theorem using this invariant. Moreover, using Weierstrass representation for maximal surfaces in Minkowski 3-space, we have classified of surfaces with cone-like singularities with some finiteness.