山田 澄生

An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring $S^1\times S^2$ and sphere $S^3$ cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number.

arXiv:1602.07072 [math.DG]

arxiv.org/abs/1711.05229

A variational formulation of Funk metric defined on a convex set in a Euclidean space is introduced. The new definition provides geometric descriptions of the Finsler metric. Secondly, the variational characterization of the Funk metric is generalized to the Weil-Petersson geometry of Teichmuller spaces. Finally, a comparison between several Funk-type metrics defined on Teichmuller spaces is made.

The goal of this paper is to introduce and to study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets of the hyperbolic space and of the sphere . We highlight some striking similarities among the three cases (Euclidean, spherical and hyperbolic) which hold at least at a formal level. The proofs of the basic properties of the classical Funk metric on subsets of use similarity properties of Euclidean triangles which of course do not hold in the non-Euclidean cases. Transforming the side lengths of triangles using hyperbolic and circular functions and using some non-Euclidean trigonometric formulae, the Euclidean similarity techniques are transported into the non-Euclidean worlds. We start by giving three representations of the Funk metric in each of the non-Euclidean cases, which parallel known representations for the Euclidean case. The non-Euclidean Funk metrics are shown to be Finslerian, and the associated Finsler norms are described. We then study their geodesics. The Hilbert geometry of convex sets in the non-Euclidean constant curvature spaces and is then developed by using the properties of the Funk metric and by introducing a non-Euclidean cross ratio. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. This leads to a formulation and a discussion of Hilbert's Problem IV in the non-Euclidean settings. Projection maps between the spaces and the upper hemisphere establish equivalences between the Hilbert geometries of convex sets in the three spaces of constant curvature, but such an equivalence does not hold for Funk geometries.

On a Teichmuller space, the Weil-Petersson metric is known to be incomplete. Taking metric and geodesic completions result in two distinct spaces, where the Hopf-Rinow theorem is no longer relevant due to the singular behavior of the Weil-Petersson metric. We construct a geodesic completion of the Teichmuller space through the formalism of Coxeter complex with the Teichmuller space as its non-linear non-homogeneous fundamental domain. We then show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-compact symmetric spaces of semi-simple Lie groups.

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.