In 2011, Chinese researcher Ni found
the solution of the Oppenheimer-Volkoff problem for a stable configuration of
stellar object with no internal source of energy. The Ni’s solution is the
nonrotating hollow sphere having not only an outer, but an inner physical
radius as well. The upper mass of the object is not constrained. In our paper,
we contribute to the description of the solution. Specifically, we give the
explicit description of metrics inside the object and attempt to link it with
that in the corresponding outer Schwarzschild solution of Einstein field
equations. This task appears to be non-trivial. We discuss the problem and
suggest a way how to achieve the continuous linkup of both object-interior and
outer-Schwarzschild metrics. Our suggestion implies an important fundamental
consequence: there is no universal relativistic speed limit, but every compact
object shapes the adjacent spacetime and this action results in the specific
speed limit for the spacetime dominated by the object. Regardless our
suggestion will definitively be proved or the successful linkup will also be
achieved in else, still unknown way, the success in the linkup represents a
constraint for the physical acceptability of the models of compact objects.

Abstract:
In this short note it is shown that all invariant metrics and functions of bounded $\mathcal C^2$-smooth domain coincide on an open non-empty subset.

Abstract:
We consider the problem of extension of pairs of continuous and bounded, partial metrics which agree on the non-empty intersections of their domains which are closed and bounded subsets of an arbitrary but fixed metric space. Two pairs of such metrics are close if their corresponding graphs are close and if the intersections of their domains are close in the Hausdorff metric. If, besides, these metrics are uniformly continuous on the intersections of their domains then there is a continuous positive homogeneous operator extending each such a pair of partial metrics to a continuous metric on the union of their domains. We prove that, in general, there is no subadditive extension operator (continuous or not) for such pairs of metrics. We provide examples showing to what extent our results are sharp and we obtain analogous results for ultrametrics.

Abstract:
The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadraic in the momenta, induced by a Killing-Stackel tensor. Our results bring to light several metrics which correspond to classically integrable dynamical systems. They include, as particular cases, the Eguchi-Hanson and Taub-NUT metrics.

Abstract:
The main result of this paper is that the space of conformally compact Einstein metrics on a given manifold is a smooth, infinite dimensional Banach manifold, provided it is non-empty, generalizing earlier work of Graham-Lee and Biquard. We also prove full boundary regularity for such metrics in dimension 4, and a local existence and uniqueness theorem for such metrics with prescribed metric and stress-energy tensor at conformal infinity, again in dimension 4. This result also holds for Lorentzian-Einstein metrics with a positive cosmological constant.

Abstract:
This article investigates the prefix vy- in so-called natural perfectives. On the basis of syntactic constructions where vy-verbs occur, it is argued that that even in natural perfectives vy- is not semantically empty.

Abstract:
We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.

Abstract:
In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is non-empty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurstons Hyperbolisation Theorem. We give examples of closed graph-manifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature.

Abstract:
On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that if non-empty the space of metrics with invertible Dirac operators is disconnected in dimensions $n \equiv 0,1,3,7 \mod 8$, $n \geq 5$. As a corollary follows results on the existence of metrics with harmonic spinors by Hitchin and B\"ar. Finally we use computations of the eta invariant by Botvinnik and Gilkey to find metrics with harmonic spinors on simply connected manifolds with a cyclic group action. In particular this applies to spheres of all dimensions $n \geq 5$.

Abstract:
This is a survey on special metrics. We shall present some results and open questions on special metrics mainly appeared in the last 10 years