Causal classification of three Misner-type spacetimes.
problem Causal structure and isocausality of three spacetimes.
method Formal proof of pairwise isocausality on covers and compactified spacetimes.
result Explicit causal bijections and deck-equivariance criterion for isocausality.
The paper classifies coverings and non-Hausdorff extensions of Misner spacetime.
problem Classifying coverings and non-Hausdorff extensions of Misner spacetime.
method Covering constructions of the punctured Minkowski plane, quotient spacetimes, and explicit embeddings.
result A natural family of spacetimes including the Hawking--Ellis extension and its universal-cover analogue.
Study explores embedding signature-changing manifolds into higher-dimensional spaces.
problem Smooth metric signature changes in spacetimes.
method Global isometric embeddings into higher-dimensional pseudo-Euclidean spaces.
result Explicit constructions of global embeddings into Minkowski and Misner spaces.
New mass definition for negative cosmological constant spacetimes.
problem Defining quasilocal mass for spacetimes with negative cosmological constant.
method Spinorial approach based on previous work for vanishing cosmological constant.
result Non-negative mass, equal to Misner-Sharp mass in spherical symmetry, zero for AdS.
In this paper, we prove Lorentzian positive mass theorem for spacetimes with distributional curvature. To do so, we introduce distributional curvature and generalized Arnowitt-Deser-Misner (ADM) momentum. As an application, we discuss a junction of spacetimes.
It is shown by several authors going back to Huisken-Yau that asymptotically Schwarzschildean time-slices possess a unique foliation by stable constant mean curvature (CMC) spheres defining the so-called CMC center of mass. We analyze how the leaves of this foliation evolve in time under the Einstein equations. More pr…
We prove that every (3+1)-dimensional flat GHMC Minkowski spacetime which is not a translation spacetime or a Misner spacetime carries a unique foliation by spacelike hypersurfaces of constant scalar curvature. In otherwords, we prove that every such spacetime carries a unique time function with isochrones of constan…
The wave equation (free boson) problem is studied from the viewpoint of the relations on the symplectic manifolds associated to the boundary induced by solutions. Unexpectedly there is still something to say on this simple, well-studied problem. In particular, boundaries which do not allow for a meaningful Hamiltonian …
We introduce a framework for studying the effects of self-interaction on the construction of point particle initial data in General Relativity. Within this framework we rigorously prove the vanishing mass claim made by Arnowitt, Deser and Misner regarding point sources. We identify a geometric structure and a scaling p…
We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation sp…
We give general sufficient conditions for the existence of trapped surfaces due to concentration of matter in spherically symmetric initial data sets satisfying the dominant energy condition. These results are novel in that they apply and are meaningful for arbitrary spacelike slices, that is they do not require any au…
Modeling wormhole creation without singularities in relativity.
problem Creating wormholes without singularities in classical relativity.
method Topological surgery and Morse theory to construct a nonsingular wormhole.
result Wormholes can be created nonsingularly in classical relativity.
In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Later, their decay assumptions were weakened by Metzger, Huang, Eichmair-M…
The (relativistic) center of mass of an asymptotically flat Riemannian manifold is often defined by certain surface integral expressions evaluated along a foliation of the manifold near infinity, e. g. by Arnowitt, Deser, and Misner (ADM). There are also what we call 'abstract' definitions of the center of mass in term…
We study the following problem: Given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be no…
The paper establishes criteria for spacetime inextendibility using asymptotic volume-distance-ratio analysis.
problem Determining inextendibility of spacetimes near singularities.
method Asymptotic analysis of volume-distance-ratio (VDR) to prove inextendibility criteria.
result Failure of VDR convergence to the Minkowski value implies inextendibility of spacetime.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in S3 and lens spaces. result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
The distance function ϱ(p,q) (or d(p,q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over Rn, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
Introduces new types of homogeneous spaces and their properties.
problem Defining and understanding new types of homogeneous spaces.
method Introducing and analyzing (strongly) (Θ-)discrete homogeneous spaces. result Discovers relationships between new and existing homogeneous space types.
This work describes compactifications of metric spaces and vector spaces using asymmetric norms.
problem Compactifying metric spaces and vector spaces using asymmetric norms.
method Nonstandard methods, ultrapowers of the spaces at hand.
result Polyhedral compactifications of vector spaces with stratified structure.
In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
problem Characterizing CAT(0) spaces with specific volume growth properties.
method Analyzing asymptotic topological regularity and volume growth.
result CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
Cantor Riemannium is a new type of space from holomorphic germs.
problem Defining a new type of space from holomorphic germs.
method Constructing the Cantor Riemannium by Borel monogenic continuation.
result The Cantor Riemannium is a metric, path connected, Gromov length space.
We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplect…
Study static Einstein-Maxwell space invariant by translation.
problem Classify static Einstein-Maxwell space invariant under translation.
method Analyze static Einstein-Maxwell space conformal to pseudo-Euclidean space.
result Complete classification of static Einstein-Maxwell space invariant under translation.
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …
The classification of G-spaces by Palais is refined for the case where the orbit space satisfies certain mild topological hypotheses. It is shown that when a sequence of such orbit spaces is "close" to a limit orbit space, in some suitable sense, within a larger ambient orbit space, the G-spaces in the tail of the sequ…
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant G-Poincaré spaces and gap conditions. result Space of isovariant structures on semifree G-Poincaré spaces is highly connected. Study shows how maps from certain geometric spaces behave near their edges.
problem Boundary regularity of harmonic maps in specific geometric spaces.
method Analysis of RCD(K,N) and CAT(0) spaces. result Established boundary regularity of harmonic maps.
Research explores the space-like embeddings in pseudo-hyperbolic space, finding geometric frames and actions.
problem Flat maximal space-like embeddings in pseudo-hyperbolic space.
method Description of Codazzi tensors, introduction of pseudo-Kähler metrics, Hamiltonian actions, moment maps, and geometric frames.
result Existence of two Hamiltonian actions with moment maps and geometric global Darboux frame.