On the differential geometric Schottky problem

利用Weierstrass表示定理，研究了n维欧几里德空间中复正交群在单连通极小曲面上的作用，并确定了作为该作用极限的极小曲面。出现极小曲面是恩内珀极小曲面的推广。如果给定的最小曲面不是全纯的，那么这个作用使最小曲面不稳定。其原因是发生了分叉（巨灾现象）。我们考虑类似Nitsch坐标系的框架（三维欧几里得空间中的曲线）的平台问题。我们通过框架的变形研究了最小曲面的分岔，并通过计算机模拟看到了突变现象的几何可视化。我们考虑由环面变形引起的环面最小曲面的变形。这与微分几何肖特基问题（研究西格尔上半空间中黎曼矩阵空间的弯曲性）密切相关。我们考虑环面变形空间的余切束中的突变集。然后得到了具有复杂结构的拉格朗日锥，并对其构造进行了推广。我们考虑了高协维欧几里得空间中Dirichlet问题（给定边界映射）的极小曲面系统。对于边界图，存在解和无解。我们可以认为在边界图的变形下出现了具有奇异点的解。实际上，Lawson和Osserman以最小锥的形式得到了一个解。我们推广了它们的构造，并给出了许多最小锥的解。此外，我们还看到最小锥具有变形空间。因此，研究包含我们的结果的最小曲面的变形是很重要的。

We consider the action of the complex orthogonal group on a simply connected minimal surface in the n-dimensional Euclidean space by using the Weierstrass representation theorem and decide the minimal surface appearing as limits of this action. An appearing minimal surface is a generalization of the Enneper minimal surface. If the given minimal surface is not holomorphic, then this action makes the minimal surface to be unstable. The reason is that the bifurcation (Catastrophe phenomena) occurs.We consider the Plateau problem for the frame (a curve in the 3 dimensional Euclidean space) like the Nitsch frame. We study the bifurcation of a minimal surface by a deformation of the frame and see the geometric visualization of the catastrophe phenomenon by a computer simulation.We consider a deformation of a minimal surface in a torus by a deformation of the torus. This is deeply relative to the Differential geometric Schottky problem (investigate the curvedness of the space of Riemann matrices in the Siegel upper half space). We consider the catastrophe set in the cotangent bundle of the space of the deformation of tori. Then we obtain a Lagrangian cone with a complex structure and generalize the construction.We consider a minimal surface system in the Euclidean space with higher co-dimension for the Dirichlet problem (a given boundary map). For the boundary map, there exists a solution and no solution.. We may consider that the solution with singularity appears under the deformation of the boundary map. In fact, Lawson and Osserman obtain a solution as a minimal cone. We generalize their construction and give many solutions as minimal cones. Moreover we see that a minimal cone have the deformation space.Hence it is important to study the deformation of minimal surfaces which contains our results.