New cosmological spacetimes without CMC Cauchy surfaces found.
problem Finding CMC Cauchy surfaces in cosmological spacetimes.
method Generalized Bartnik's construction to connected sums of three-manifolds.
result Cosmological spacetimes without CMC Cauchy surfaces for any compact three-manifolds.
Solves a Cauchy problem for minimal spacelike surfaces in 4D spacetime.
problem Constructing minimal spacelike surfaces in 4D spacetime.
method Defining isoclinic parametric surfaces and proving their relation to holomorphic functions, solving the Cauchy problem.
result Solves the Cauchy problem for minimal spacelike surfaces in R24. We establish a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. Some of the basic results in Thurston's theory of measured laminations on surfaces are derived from the Cauchy inequality.
Proves symmetry in vacuum spacetimes with compact Cauchy horizons.
problem Symmetries in vacuum spacetimes with compact Cauchy horizons.
method Solving Killing equation up to infinite order at the Cauchy horizon, extending the solution to the globally hyperbolic region.
result Maximally globally hyperbolic vacuum development cannot be extended across compact Cauchy horizons.
Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.
problem Embedding cone-metrics in anti-de Sitter spacetimes.
method Proving embeddings using Fuchsian representations and GHMC spacetimes.
result Unique embeddings of cone-metrics in GHMC anti-de Sitter spacetimes.
New vacuum spacetimes without CMC Cauchy surfaces found.
problem Finding vacuum cosmological spacetimes without CMC Cauchy surfaces.
method Extended construction of [6] using spatial topologies M#M. result Obtained a large class of vacuum cosmological spacetimes.
New spacetimes without CMC Cauchy surfaces are shown to be incomplete.
problem Vacuum cosmological spacetimes without constant mean curvature Cauchy surfaces.
method Construction and analysis of specific spacetimes with large gluing parameters.
result These spacetimes are shown to be null geodesically incomplete.
Study shows existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems.
problem Existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems of Novikov equation.
method Kato's semi-group approach to establish well-posedness and C1 regularity. result Existence of exactly two families of periodic one-forms defining metrics and Levi-Civita connections.
We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature S1×S2 Cauchy surface also contains a maximal Cauchy surface. Combining …
We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.
We consider globally hyperbolic spacetimes with compact Cauchy surfaces in a setting compatible with the presence of a positive cosmological constant. More specifically, for 3+1 dimensional spacetimes which satisfy the null energy condition and contain a future expanding compact Cauchy surface, we establish a precise c…
Study wave equations on compact Cauchy horizons with unique solutions.
problem Existence and uniqueness of solutions to wave equations near compact Cauchy horizons.
method Energy estimates and Cauchy horizon analysis for wave equations on vector bundles.
result Unique solution to wave equations on globally hyperbolic region if initial data is given on Cauchy horizon.
New CMC existence result for expanding cosmological spacetimes.
problem Establishing a new constant mean curvature (CMC) existence result for cosmological spacetimes.
method Construction of barriers in the support sense and asymptotic limit of mean curvature flow.
result The existence of a CMC Cauchy surface in expanding cosmological spacetimes.
Extending BTZ models to complete hyperbolic surfaces.
problem Extending BTZ models to complete hyperbolic surfaces.
method Proving a parametrization result for globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
result The tangent bundle of the Teichmüller space parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
New method constructs spacelike data leading to trapped surfaces.
problem Formation of trapped surfaces from spacelike initial data.
method Free data formalism and local existence result.
result Data can be extended to asymptotically flat Cauchy data.
Proves compact Cauchy horizons have constant surface gravity under null energy condition.
problem Proving compact Cauchy horizons have constant surface gravity.
method Combines ergodic theory, Hodge theory, and Riemannian flow theory.
result Compact Cauchy horizons admit a smooth lightlike tangent vector field of constant surface gravity.
Lorentzian distances to Cauchy surfaces fail to be locally equi-Lipschitz.
problem Lorentzian distances to Cauchy surfaces
method Conjectures based on Cauchy temporal functions
result Lorentz distances to Cauchy surfaces are not locally equi-Lipschitz
Revisits equations for pseudospherical surfaces, linking historical and current research.
problem Understanding equations for pseudospherical surfaces.
method Historical review and current research topics.
result Current research connects pseudospherical surfaces to Cauchy problems.
Proves existence of Killing fields in smooth spacetimes with compact Cauchy horizons.
problem Existence of Killing fields in smooth spacetimes with compact Cauchy horizons.
method Normalized surface gravity of compact non-degenerate Cauchy horizons in smooth vacuum spacetimes.
result Proves the Isenberg-Moncrief conjecture on the existence of Killing fields.
Geometric approach to Dirac operator evolution on spacetimes.
problem Constructing the Cauchy evolution operator for Lorentzian Dirac operators.
method Realizing the operator as a sum of oscillatory integrals, relating to Feynman propagator.
result Relating Cauchy evolution operators to Feynman propagators and constructing Hadamard states.
Formula calculates index for CR operators on surfaces with boundary punctures.
problem Computing the index for Cauchy-Riemann operators on surfaces with boundary punctures.
method Large antilinear deformations method, generalized to punctured surfaces.
result Involves a non-standard weighted count of boundary zeros in the Euler characteristic term.
Paper establishes equivalence between algebraic and functorial QFTs.
problem Challenges in globally hyperbolic Lorentzian bordisms and their impact on field theories.
method Introduces a new concept of bordisms and pseudo-operads to address challenges, defining FQFTs as pseudo-multifunctors.
result Equivalence theorem between globally hyperbolic Lorentzian FQFTs and AQFTs.
The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.
problem Counting pseudo-holomorphic curves in symplectic Calabi-Yau 3-folds.
method Constructs three chambered invariants: nBl, n1,2, n2,1, defined by counting solutions to ADHM vortex equations and pseudo-holomorphic sections of bundles. result Conjectures a relationship between n1,2 and n2,1 and symplectic invariants. We give a construction that connects the Cauchy problem for Liouville elliptic equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H3, and solve both of them. We construct the only mean curvature one surface in H3 that passes through a given curve with given unit normal a…
The zoology of singularities for Lorentzian manifold is slightly more complicated than for Riemannian manifolds. Our present work study Cauchy-compact globally hyperbolic singular flat spacetimes with extreme BTZ-like singular lines. We use the notion of BTZ-extension of a singular spacetime introduced in a previous pa…
Flat metrics on hyperbolic surfaces embed as polyhedral surfaces in (2+1)-spacetimes.
problem Embedding flat metrics on hyperbolic surfaces into (2+1)-spacetimes.
method Using convex polyhedral Cauchy surfaces and Teichmüller space properties.
result Existence and uniqueness of flat metrics embedding in (2+1)-spacetimes.
No trapped surfaces can form under low-regularity bounds in certain spacetimes.
problem Existence of trapped surfaces in low regularity solutions to Einstein's equations.
method Analyzing the initial data in Besov B2,13/2 norm and extending to H3/2 smallness. result No trapped surfaces can exist initially when the Cauchy data are close to Minkowski spacetime data.
The group of conformal diffeomorphisms and the group of causal automorphisms on two-dimensional globally hyperbolic spacetimes are clarified. It is shown that if spacetimes have non-compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Minkowski spacetime, and if spacetimes have compact Cauc…
Chernov-Nemirovski observed that the existence of a globally hyperbolic Lorentzian metric on a (3 + 1)-spacetime pins down a smooth structure on the underlying 4-manifold. In this paper, we point out that the diffeomorphism type of a globally hyperbolic (n + 1)-spacetime is determined by the h-cobordism class of its Ca…
We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surfa…
We consider a connection ∇X on a complex line bundle over a Riemann surface with boundary M0, with connection 1-form X. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) L:=∇X∗∇X+q, with q a complex valued potential, uniquely determines the…
Researchers create initial data for multiple collapsing boson stars.
problem Forming multiple trapped surfaces in spacetime.
method Constructed Cauchy initial data for EMKG system.
result Multiple trapped surfaces form in finite time.
We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and use these solutions to compute several new examples. We give the criteria on the …
The paper discusses when spacetimes have constant mean curvature slices, an important but unknown condition.
problem The existence of constant mean curvature (CMC) Cauchy surfaces in spacetimes is crucial but not always guaranteed.
method Expository review of existing results and conjectures about CMC slices in spacetimes.
result It is not known whether spacetimes with CMC slices are generic, highlighting an important open problem.
Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.
problem Analyzing the relationship between Lorentz harmonic maps and spacelike surfaces.
method Using loop group techniques, develop DPW-type representations and solve Cauchy problems.
result Establish a correspondence between Lorentz harmonic maps and spacelike immersions, leading to families of surfaces of constant Gauss curvature.
In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.
In this paper we study the flat (n+1)-spacetimes admitting a Cauchy surface diffeomorphic to a compact hyperbolic n-manifold. We show how to construct a canonical future complete one among all such spacetimes sharing the same holonomy. We study the geometry of such a spacetime in terms of its canonical cosmological tim…
When studying the causal propagation of a field in a globally hyperbolic spacetime M, one often wants to express the physical intuition that it has compact support in spacelike directions, or that its support is a spacelike compact set. We compare a number of logically distinct formulations of this idea, and of the com…
Properties of the Cauchy-Riemann-Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3-surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions …
Study Schiffer operators on Riemann surfaces, linking conformal and topological invariants.
problem Investigate Schiffer operators on Riemann surfaces and their connections to conformal and topological invariants.
method Develop calculus for Schiffer and Cauchy operators, derive index theorems, and characterize kernels and images.
result Derive index theorems for Schiffer operators, connecting conformal invariants to topological invariants.
We prove identification of coefficients up to gauge by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of C. In the geometric setting, we fix a Riemann surface with boundary, and consider both a Dirac-type operator plus potential acting on sections of a …
We solve the Cauchy-Dirichlet problem for the minimal surface system in arbitrary dimension and codimension assuming a condition on the variation of the initial submanifold .
The Degasperis-Procesi equation's solutions define pseudospherical metrics and can lead to surface collapse.
problem Understanding the breakdown of manifolds determined by Cauchy problems of the Degasperis-Procesi equation.
method Analyzing the pseudospherical nature of local and non-local formulations of the Degasperis-Procesi equation.
result Solutions to Cauchy problems with non-trivial initial data define an orthonormal coframe for pseudospherical metrics.
Constructs all real analytic germs of zero mean curvature surfaces in Lorentz-Minkowski 3-space.
problem Analyzing surfaces with light-like points in Lorentz-Minkowski 3-space.
method Applying the Cauchy-Kovalevski theorem for partial differential equations.
result Surfaces with light-like points in Lorentz-Minkowski 3-space contain a light-like line when they do not change causal types.
Investigates parallel spinors on Lorentzian four-manifolds using differential geometry.
problem Characterizing and classifying Lorentzian four-manifolds with parallel spinors.
method Formulated parallel spinor flow equations and used parabolic pairs theory.
result Characterized all parallel Cauchy pairs on simply connected Cauchy surfaces and classified compact three-manifolds.
This paper has been withdrawn by the authors because it has been merged with paper arXiv:0903.3501v1 [math.DG]
We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral approximation.
New data shows multiple black holes can form from modified vacuum data.
problem Formation of multiple black holes from modified vacuum data.
method Modified Brill-Lindquist metric, short-pulse framework, stability analysis, gluing method.
result Formation of multiple black holes in finite time under weak cosmic censorship.