山田 澄生

We present several new space-periodic solutions of the static vacuum Einstein equations in higher dimensions, both with and without black holes, having Kasner asymptotics. These latter solutions are referred to as gravitational solitons. Further partially compactified solutions are also obtained by taking appropriate quotients, and the topologies are computed explicitly in terms of connected sums of products of spheres. In addition, it is shown that there is a correspondence, via Wick rotation, between the spacelike slices of the solitons and black hole solutions in one dimension less. As a corollary, the solitons give rise to complete Ricci flat Riemannian manifolds of infinite topological type and generic holonomy, in dimensions 4 and higher.

We construct a new set of asymptotically flat, static vacuum solutions to the Einstein equations in dimensions 4 and 5, which may be interpreted as a superposition of positive and negative mass black holes. The resulting spacetimes are axisymmetric in 4-dimensions and bi-axisymmetric in 5-dimensions, and are regular away from the negative mass singularities, for instance conical singularities are absent along the axes. In 5-dimensions, the topologies of signed mass black holes used in the construction may be either spheres $S^3$ or rings $S^1 \times S^2$; in particular, the negative mass static black ring solution is introduced. A primary observation that facilitates the superposition is the fact that, in Weyl-Papapetrou coordinates, negative mass singularities arise as overlapping singular support for a particular type of Green's function. Furthermore, a careful analysis of conical singularities along axes is performed, and formulas are obtained for their propagation across horizons, negative mass singularities, and corners. The methods are robust, and may be used to construct a multitude of further examples. Lastly, we show that balancing does not occur between any two signed mass black holes of the type studied here in 4 dimensions, while in 5 dimensions two-body balancing is possible.

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]

https://arxiv.org/abs/1807.03452<br /> <br /> To appear in Trans. AMS

arxiv.org/abs/1711.05229

https://arxiv.org/abs/1805.11385

Chapter 4 in Riemann to Geometry and Rel-<br /> ativity edited by L. Ji, A. Papadopoulos and S. Yamada Springer (2017)

For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, we exhibit new geodesics for the arc metric on the Teichmuller space of an arbitrary surface of negative Euler characteristic with nonempty boundary. We also obtain new geodesics for Thurston's metric on Teichmuller spaces of hyperbolic surfaces without boundary.

In this paper we investigate the extension of the charged Riemannian Penrose inequality to the case where charges are present outside the horizon. We prove a positive result when the charge densities are compactly supported, and present a counterexample when the charges extend to infinity. We also discuss additional extensions to other matter models.

We present the outline of a proof of the Riemannian Penrose inequality with charge r &lt;= m + root m(2) - q(2), where A = 4 pi r(2) is the area of the outermost apparent horizon with possibly multiple connected components, m is the total ADM mass, and q the total charge of a strongly asymptotically flat initial data set for the Einstein-Maxwell equations, satisfying the charged dominant energy condition, with no charged matter outside the horizon.

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.

The most general formulation of Penrose's inequality yields a lower bound for Arnowitt-Deser-Misner mass in terms of the area, charge, and angular momentum of black holes. This inequality is in turn equivalent to an upper and lower bound for the area in terms of the remaining quantities. In this paper, we establish the lower bound for a single black hole in the setting of axisymmetric maximal initial data sets for the Einstein-Maxwell equations, when the non-electromagnetic matter fields are not charged and satisfy the dominant energy condition. It is shown that the inequality is saturated if and only if the initial data arise from the extreme Kerr-Newman spacetime. Further refinements are given when either charge or angular momentum vanish. Last, we discuss the validity of the lower bound in the presence of multiple black holes.

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in R-n, and investigate its properties. Two guiding principles are given by Milnor's way of measuring using a local Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature functional is that it allows us to interpret the singular/noneuclidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, so that one can compute the total curvature for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the curvature. A collection of results on upper/lower bounds of the total curvature on isotopy/homeomorphism classes of embeddings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional measuring complexity of spatial graphs in differential-geometric terms.

We continue the study of minimal singular surfaces obtained by a minimization of a weighted energy functional in the spirit of J. Douglas's approach to the Plateau problem. Modeling soap films spanning wire frames, a singular surface is the union of three disk-type surfaces meeting along a curve which we call the free boundary. We obtain an a priori regularity result concerning the real analyticity of the free boundary curve. Using the free boundary regularity of the harmonic map, we construct local harmonic isothermal coordinates for the minimal singular surface in a neighborhood of a point on the free boundary. Applications of the local uniformization are discussed in relation to H. Lewy's real analytic extension of minimal surfaces.

We show that the Brill-Lindquist initial data provide a counterexample to a Riemannian Penrose inequality with charge conjectured by Gibbons. The observation illustrates a sub-additive characteristic of the area radii for the individual connected components of an outermost horizon as a lower bound of the ADM mass.

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.

We reinterpret the proof of the Riemannian Penrose inequality by Bray. The modified argument turns out to have a nice feature so that the flow of Riemannian metrics appearing in Bray&apos;s proof gives a Lorentzian metric of a spacetime. We also discuss a possible extension of our approach to charged black holes.

It is expected that matter composed of a perfect fluid cannot be at rest outside of a black hole if the spacetime is asymptotically flat and static (nonrotating). However, there has not been rigorous proof for this expectation without assuming spherical symmetry. In this paper, we provide a proof of nonexistence of matter composed of a perfect fluid in static black hole spacetimes under certain conditions, which can be interpreted as a relation between the stellar mass and the black hole mass. (c) 2006 American Institute of Physics.

For a given boundary Gamma in R-n consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface Sigma spanning Gamma. Sigma is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Gamma. We introduce a notion of total curvature C-tot(Gamma) for such graphs, or nets, Gamma. We show that 2 pi times the area density of Sigma at any point is less than or equal to C-tot(Gamma). For n = 3, these density estimates imply, for example, that if C-tot(Gamma) &lt;= 3.649 pi, then the only possible singularities of a piecewise smooth (M, 0, delta)-minimizing set Sigma are curves, along which three smooth sheets of Sigma meet with equal angles of 120 degrees.