山田 澄生

The Steiner problem is the problem of finding the shortest network connecting a given set of points. By the singular Plateau Problem, we will mean the problem of finding an area-minimizing surface ( or a set of surfaces adjoined so that it is homeomorphic to a 2-complex) spanning a graph. In this paper, we study the parametric versions of the Steiner problem and the singular Plateau problem by a variational method using a modified energy functional for maps. The main results are that the solutions of our one- and two-dimensional variational problems yield length and area minimizing maps respectively, i.e. we provide new methods to solve the Steiner and singular Plateau problems by the use of energy functionals. Furthermore, we show that these solutions satisfy a natural balancing condition along its singular sets. The key issue involved in the two-dimensional problem is the understanding of the moduli space of conformal structures on a 2-complex.

We construct a time-symmetric asymptotically flat initial data set to the Einstein-Maxwell Equations which satisfies m - 1/2 (R + O-2/R) < 0 where m is the total mass, R = √ A/4π is the area radius of the outermost horizon and Q is the total charge. This yields a counter-example to a natural extension of the Penrose Inequality for charged black holes.

Given a surface of higher genus, we will look at the Weil-Petersson completion of the Teichmuller space of the surface, and will study the geometry induced by the Weil-Petersson distance functional. Although the completion is no longer a Riemannian manifold, it has characteristics similar to those of Cartan-Hadamard manifolds.

The well-known monotonicity formula for harmonic maps says that the scaled energy functional over a ball of radius r is a non-decreasing function of r. The proof uses the fact that the energy functional is critical under any compactly supported variation on the domain of the map. In this article, we will instead use the fact that the energy is critical under variations of the map on the image of the map. By choosing the variational vector field suitably it will be shown that a scaled energy considered as an integral functional over a ball of radius r where r is the distance from a point on the image manifold, is monotonically non-decreasing. The formula takes a stronger form when the image is one dimensional.