arXiv research

Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.

This study finds a special class of representations that dominate others in a complex hyperbolic group.

Study convex hyperbolic cone-metrics on 3-manifold boundaries, proving unique bent realizations.

New cosmological spacetimes without CMC Cauchy surfaces found.

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.

New vacuum spacetimes without CMC Cauchy surfaces found.

Study shows existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems.

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 $S^1 \times S^2$ 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 prove that any smooth vacuum spacetime containing a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant admits a Killing vector field. This proves a conjecture by Moncrief and Isenberg from 1983 under the assumption on the surface gravity and generalises previous results due to …

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…

New CMC existence result for expanding cosmological spacetimes.

Extending BTZ models to complete hyperbolic surfaces.

New method constructs spacelike data leading to trapped surfaces.

Proves compact Cauchy horizons have constant surface gravity under null energy condition.

Revisits equations for pseudospherical surfaces, linking historical and current research.

Lorentzian distances to Cauchy surfaces fail to be locally equi-Lipschitz.

Proves existence of Killing fields in smooth spacetimes with compact Cauchy horizons.

Geometric approach to Dirac operator evolution on spacetimes.

We consider two random group models: the hexagonal model and the square model, defined as the quotient of a free group by a random set of reduced words of length four and six respectively. Our first main result is that in this model there exists a sharp density threshold for Kazhdan's Property (T) and it equals 1/3. Ou…

Formula calculates index for CR operators on surfaces with boundary punctures.

Paper establishes equivalence between algebraic and functorial QFTs.

The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.

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.

No trapped surfaces can form under low-regularity bounds in certain spacetimes.

Chruściel, Isenberg, and Pollack constructed a class of vacuum cosmological spacetimes that do not admit Cauchy surfaces with constant mean curvature. We prove that, for sufficiently large values of the gluing parameter, these examples are both future and past null geodesically incomplete. The authors are honored to de…

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…

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…

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 give a definition of isoclinic parametric surfaces in $\mathbb{R}^4_2$ and prove that such an isoclinic conformal immersion comes from two holomorphic functions. A Cauchy problem was proposed and solved, namely: construct an isoclinic and minimal positive (negative) spacelike surface in $\mathbb{R}^4_2$, containing …

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 $\nabla^X$ on a complex line bundle over a Riemann surface with boundary $M_0$, with connection 1-form $X$. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) $L:={\nabla^X}^*\nabla^X + q$, with $q$ a complex valued potential, uniquely determines the…

Researchers create initial data for multiple collapsing boson stars.

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 …

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.

Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.

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…

The paper generalizes the second Pappus-Guldin theorem for calculating volumes of bodies.

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 …

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…

Study Schiffer operators on Riemann surfaces, linking conformal and topological invariants.

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 .

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 $\mathbb{C}$. 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 …

The Degasperis-Procesi equation's solutions define pseudospherical metrics and can lead to surface collapse.

Investigates parallel spinors on Lorentzian four-manifolds using differential geometry.

This paper has been withdrawn by the authors because it has been merged with paper arXiv:0903.3501v1 [math.DG]