The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method's construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
Trend · papers per month
48 results for “CMC slicings”
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.
The paper studies CMC foliations and their conformal aspects on Riemannian manifolds.
problem Understanding and normalizing CMC foliations on conformally compact manifolds.
method Non-linear PDEs and conformal transformations.
result Locally, any slicing can be made into a CMC foliation by conformal changes.
Study describes bifurcations and solutions of a specific equation on a particular manifold.
problem Analyzing bifurcations and solutions of the vacuum Lichnerowicz equation.
method Exhaustive description of bifurcations and solutions on with invariant seed data.
result Description of CMC slicings of Schwarzschild-de Sitter and Nariai solutions.
Study finds existence and uniqueness of CMC graphs in multiply connected domains.
problem Existence and uniqueness of constant mean curvature graphs in multiply connected domains.
method Established through rigorous mathematical analysis and proof.
result Existence and uniqueness of compact graphs of constant mean curvature.
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…
Study of hypersurfaces in static spacetimes with curvature bounds.
problem Characterizing hypersurfaces in static spacetimes with curvature constraints.
method Mean curvature and gradient estimates for spacelike hypersurfaces.
result Complete CMC hypersurfaces in static spacetimes are maximal under certain curvature conditions.
This work characterizes asymptotically hyperbolic 3-manifolds via CMC-foliations.
problem Characterizing asymptotically hyperbolic 3-manifolds.
method Existence of a suitable CMC-foliation with geometric curvature estimates.
result Any 3-manifold is asymptotically hyperbolic if it has a CMC-cover with controlled instability.
We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minko…
Constructs constant mean curvature foliations in Schwarzschild spacetime.
problem Finding constant mean curvature foliations in Schwarzschild spacetime.
method Constructs -axisymmetric, spacelike, spherically symmetric, constant mean curvature hypersurfaces.
result Mean curvature varies in each slice and ranges from minus infinity to plus infinity.
We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the ex…
We will discuss some sharp estimates for CMC graphs in a Riemannian 3-manifold MxR whose boundary is contained in a slice. We will start by giving sharp lower bounds for the geodesic curvature of the boundary and improve these bounds when assuming additional restrictions on the maximum height that such a surface reache…
The paper constructs unstable and outlying CMC spheres in specific manifolds.
problem Stability of CMC spheres in certain manifolds.
method Non-linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry.
result Existence of unstable CMC spheres, indicating the stability condition cannot be removed.
New CMC surfaces with dihedral symmetry constructed from Darboux transforms.
problem Constructing closed CMC surfaces with specific symmetries.
method Using Darboux transforms on multiple covers to create new CMC surfaces.
result Explicit parametrisations of new closed CMC surfaces with dihedral symmetry.
No proper biharmonic CMC compact hypersurface in a specific warped product space.
problem Existence of proper biharmonic CMC hypersurfaces in a warped product space.
method Finding necessary and sufficient conditions for proper biharmonic CMC hypersurfaces in a special warped product space.
result No proper biharmonic CMC compact hypersurface exists in the specified space.
We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loo…
Proves non-vanishing CMC flux for specific manifolds.
problem Proving non-vanishing CMC flux for certain manifolds.
method Analyzes Riemannian manifolds with constant mean curvature.
result Non-vanishing CMC flux for specified manifolds.
Constructs cmc doublings of minimal surfaces via min-max theory.
problem Construct cmc doublings of minimal surfaces.
method Uses min-max theory and catenoid estimate.
result Constructs ε-cmc doublings of Σ for small ε > 0.
We translate a classification scheme for periodic CMC surfaces developed by J. Dorfmeister and the author to discrete CMC surfaces in the sense of A. Bobenko and U. Pinkall. The scheme uses the dressing action on discrete CMC surfaces to arrive at a classification for periodic discrete CMC surfaces.
Compactness proven for CMC surfaces with bounded topology and boundary length.
problem Proving compactness of CMC surfaces with specific constraints.
method Graphical compactness proof for surfaces with bounded topology, area, and boundary length.
result Space of free boundary CMC surfaces is compact in the graphical sense away from a finite set of points.
Proves special foliation property of SS-CMC hypersurfaces in Schwarzschild spacetime.
problem Characterizing and foliating SS-CMC hypersurfaces in Schwarzschild spacetime.
method Characterization and proof of foliation property using SS-CMC hypersurfaces.
result Verifies part of Malec and Ó Murchadha's conjecture.
New theorem shows singularities in high-density cosmologies without global assumptions.
problem Proving singularities in cosmological models without global topological constraints.
method Past null focusing condition and Einstein field equations.
result All timelike geodesics are past incomplete in high-density scenarios.
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.
Unique CMC foliation in Minkowski space solved.
problem Classifying entire spacelike CMC hypersurfaces in Minkowski space.
method Proved uniqueness of spacelike CMC foliation for regular domains.
result Uniquely foliated by spacelike CMC hypersurfaces for any regular domain.
Study singularities of CMC 1 surfaces in de Sitter space.
problem Characterize singularities of spacelike CMC 1 surfaces in de Sitter space.
method Proved duality between singular points and introduced invariants.
result Classified non-degenerate singular points on spacelike CMC 1 surfaces.
Survey on CMC surfaces in homogeneous 3-manifolds.
problem Existence and descriptive results for CMC surfaces.
method Survey of recent developments in CMC surfaces theory.
result Existence and descriptive results for CMC laminations and foliations.
Proves existence and uniqueness of CMC foliations in cuspidal manifolds.
problem Existence and uniqueness of CMC foliations in asymptotically cuspidal manifolds.
method Proof without curvature assumptions, applicable in any dimension.
result Proves existence and uniqueness of CMC foliations.
Estimates bandwidth for CMC initial data sets.
problem Estimating bandwidth for constant mean curvature (CMC) initial data sets.
method Three independent proofs: stability of null expansion, spacetime harmonic function perturbation, Dirac operator.
result Generalized Gromov's band width estimate to CMC initial data sets.
Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces established.
problem Establishing curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces.
method Pointwise curvature estimate and sheeting theorem for weakly stable CMC hypersurfaces.
result Effective version of the compactness theorem for weakly stable CMC hypersurfaces.
The paper explores CMC hypersurfaces in spheres, verifying Yau's conjecture.
problem Exploring the space of CMC hypersurfaces in spheres.
method Description and verification of CMC hypersurfaces, focusing on cases.
result Verification of Yau's conjecture for minimal hypersurfaces in spheres.
Authors review CMC spacelike hypersurfaces and new existence results.
problem Existence of constant mean curvature (CMC) spacelike hypersurfaces in cosmological spacetimes.
method Review and new existence results based on previous work and conjectures.
result New existence results for CMC spacelike hypersurfaces.
Study finds existence and non-existence of large stable CMC spheres in asymptotically flat 3-manifolds.
problem Existence and non-existence of large stable CMC spheres in asymptotically flat 3-manifolds.
method Extends Lyapunov-Schmidt analysis to 'far-off-center' regime and general Schwarzschild asymptotics.
result Sharp existence and non-existence results for large stable CMC spheres.
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.
Python tool for constructing CMC surfaces using DPW method.
problem Constructing minimal and constant mean curvature surfaces.
method DPW method implemented in Python.
result Python package for CMC surfaces in various space forms.
In this paper we numerically construct CMC deformations of the Lawson minimal surfaces using a spectral curve and a DPW approach to CMC surfaces in spaceforms.
Study of CMC spheres in Heisenberg group, proving conjecture.
problem Proving conjecture about CMC spheres in Heisenberg group.
method Analyzing properties of spheres in Riemannian Heisenberg group .
result Supporting conjecture that CMC spheres are isoperimetric sets.
Biharmonic tori in spheresmath.DG
Study on proper-biharmonic flat tori in spheres with CMC conditions.
problem Finding conditions for CMC proper-biharmonic immersions of tori in spheres.
method Analyzing rectangular and square tori, finding necessary and sufficient conditions, and explicit expressions.
result Explicit expressions of some CMC proper-biharmonic immersions of certain tori in spheres.
The study constructs free boundary CMC annuli in spherical and hyperbolic balls.
problem Finding free boundary CMC annuli in spherical and hyperbolic balls.
method Constructing free boundary CMC annuli with constant mean curvature H in geodesic balls of S^3 and H^3.
result Embedded free boundary CMC annuli exist for certain mean curvatures in both spaces.
Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.
problem Understanding CMC hypersurfaces with bounded index and area.
method Bubble-compactness theory for embedded CMC hypersurfaces in low dimensions.
result Minimal blow-ups are all catenoids, and bounds on genus provided.
CMC hierarchymath.DG
We propose an extension of the differential system for constant mean curvature (CMC) surfaces in a three dimensional space form to an associated hierarchy of evolution equations by the higher-order commuting symmetries. The infinite sequence of higher-order conservation laws of CMC surfaces admit the corresponding exte…
It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.
The paper proves that spacelike CMC surfaces cannot have fold singular points.
problem Investigating fold singular points on spacelike CMC surfaces.
method Analytical proof and duality analysis between conelike and fold singular points.
result Spacelike CMC surfaces do not admit fold singular points.
We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…
The paper constructs all cmc hypersurfaces with two principal curvatures.
problem Finding all hypersurfaces with constant mean curvature and two principal curvatures.
method Explicit immersions and parameter analysis for hypersurfaces in space forms.
result The family of cmc hypersurfaces with two principal curvatures depends on two parameters, H and C.
The abstract proves the existence of CMC-1 surfaces with any complex structure in hyperbolic space.
problem Proving the existence of CMC-1 surfaces with arbitrary complex structures in hyperbolic space.
method Using a jet interpolation theorem and a uniform approximation theorem for holomorphic null curves.
result Existence of complete densely immersed CMC-1 surfaces in hyperbolic space with arbitrary complex structure.
This paper establishes an interesting connection between the family of CMC surfaces of revolution in and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in the , only spacelike cylinders and stand…
The paper proves properties of triharmonic CMC hypersurfaces with specific curvature conditions.
problem Characterizing triharmonic CMC hypersurfaces with distinct principal curvatures.
method Analyzing critical points of the tri-energy and applying geometric properties.
result Proves conditions for constant scalar curvature and minimality of hypersurfaces.
The paper proves existence of multiple closed CMC hypersurfaces with small mean curvature.
problem Existence of multiple closed CMC hypersurfaces with small mean curvature.
method Analyzes a closed Riemannian manifold to prove the existence of multiple closed CMC hypersurfaces with optimal regularity.
result For all , there exists such that if , contains at least many closed -CMC hypersurfaces with optimal regularity.