Researchers construct a gauge-invariant energy functional for axially symmetric perturbations around Kerr black holes.
problem Understanding energy of axially symmetric perturbations around Kerr black holes.
method Hamiltonian dimensional reduction to a 2+1 Einstein-wave map system, constructing a positive-definite, gauge-invariant energy functional. result The energy functional serves as a Hamiltonian for the constrained evolution of linear perturbations.
This work proves Kerr black holes are dynamically stable under certain perturbations.
problem Dynamical stability of Kerr black holes under axially symmetric perturbations.
method Dimensional reduction to 2+1 Einstein-wave map system, construction of positive-definite energy functional, proving boundary terms vanish.
result Strictly conserved positive energy for axially symmetric linear perturbations of Kerr black holes.
Researchers found a way to measure energy in black hole perturbations.
problem Lack of positive-definite and conserved energy in black hole stability.
method Dimensional reduction and construction of a positive-definite energy functional.
result Conserved Hamiltonian energy for axially symmetric perturbations of Kerr black holes.
This paper tackles gauge fixing and regularity for perturbations around spherical backgrounds.
problem Understanding gauge freedom and regularity in perturbation theory for symmetric tensors.
method Analyzing Hodge-type decomposition for axially symmetric and axistationary tensors, showing existence and uniqueness of gauge tensors.
result Stationary and axially symmetric second order perturbations can be rendered in a canonical form with only one degree of differentiability loss near the origin.
Study shows stability of Schwarzschild spacetime under specific perturbations.
problem Linear stability of Schwarzschild spacetime under axial perturbations.
method Complex line bundle interpretation and connection-level object analysis.
result Suitably regular initial data decay to a linearized Kerr metric.
We discuss our recent work [4] in which gravitational radiation was studied by evaluating the Wang-Yau quasi-local mass of surfaces of fixed size at the infinity of both axial and polar perturbations of the Schwarzschild spacetime, à la Chandrasekhar [1].
Stability of Schwarzschild singularity in near-Schwarzschild black holes under perturbations.
problem Stability of the Schwarzschild singularity in near-Schwarzschild black holes.
method Energy methods and new approach to Einstein vacuum equations in axial symmetry.
result The solution displays asymptocially-velocity-term-dominated dynamics and approaches a different Kasner solution at each point of the singularity.
We prove that Wilson loop expectation values for arbitrary simple closed contours obey an area law up to second order in perturbative two-dimensional Yang-Mills theory. Our analysis occurs within a general family of axial-like gauges, which include and interpolate between holomorphic gauge and the Wu-Mandelstam-Liebran…
The standard Feynman diagrammatic approach to quantum field theories assumes that perturbation theory approximates the full quantum theory at small coupling even when a mathematically rigorous construction of the latter is absent. On the other hand, two-dimensional Yang-Mills theory is a rare (if not the only) example …
New surfaces near a sphere violate Minkowski inequality.
problem Minkowski inequality failure near a sphere.
method Constructed surfaces converging to a sphere in W2,p∩C1. result Minkowski inequality fails for perturbations of a sphere.
Derives equations for gravitational and electromagnetic perturbations of Reissner-Nordström spacetime.
problem Global non-linear stability of Reissner-Nordström spacetime under polarized perturbations.
method Derives a system of equations through a Chandrasekhar-type transformation.
result Derives a gauge invariant quantity associated to the electromagnetic tensor that verifies a Regge-Wheeler equation.
The paper examines the stability of Minkowski inequality for nearly spherical domains.
problem Stability of Minkowski inequality for nearly spherical domains.
method Analyzes stability inequalities for C1 perturbations of a ball and axially symmetric perturbations. result Established stability inequalities for curvature integrals of nearly spherical domains.
We show that a stationary asymptotically flat electro-vacuum solution of Einstein's equations that is everywhere locally "almost isometric" to a Kerr-Newman solution cannot admit more than one event horizon. Axial symmetry is not assumed. In particular this implies that the assumption of a single event horizon in Alexa…
New curvature defined for corank 1 singular surfaces in 3D.
problem Defining curvature for singular surfaces.
method Introducing axial vector and curvature parabola to define axial curvature.
result Relates axial curvature to Gaussian curvature of a blow-up for certain singularities.
The article improves Beckner's inequality for axially symmetric functions on the n-dimensional sphere.
problem Improving Beckner's inequality for axially symmetric functions on Sn. method Uniqueness and existence results for Q-curvature type equations with a Paneitz operator on Sn for axially symmetric functions. result Improved Beckner's inequality for axially symmetric functions on Sn. Study on axially symmetric surfaces' flow, showing all singularities are of type I.
problem Understanding singularity formation in axially symmetric mean curvature flow.
method Analysis of Neumann boundary conditions and type of singularities.
result All singularities at first time are of type I.
New method constructs axial vector fields and defines quasi-local spin-angular momentum.
problem Constructing axial vector fields on Riemannian two-spheres.
method Using centre-of-mass unit sphere reference systems and Lie-propagated unit sphere reference systems.
result Constructive definition of quasi-local spin-angular momentum and balance relations.
Improved Beckner's inequality for axially symmetric functions on S^4.
problem Proving axially symmetric solutions to a constant Q-curvature type equation must be constant.
method Analyzing constant Q-curvature type equations on S^4, using Pohozaev-type identities and bifurcation methods.
result Improved Beckner's inequality for axially symmetric functions on S^4.
New surfaces described that are symmetric and solve a specific equation.
problem Understanding symmetric shapes of membranes.
method Characterized and described axially symmetric Helfrich spheres using the reduced membrane equation.
result These surfaces are symmetric and belong to a specific family.
The paper proves and analyzes Minkowski inequalities for nearly spherical domains.
problem Validating and stabilizing Minkowski inequalities for perturbed balls.
method Analyzing C1-perturbations of the ball, proving sharp and almost sharp inequalities. result Sharp geometric and almost sharp Minkowski inequalities for nearly spherical domains.
Finite index solutions to Bernoulli problem are always axially symmetric.
problem Entire solutions to the Bernoulli free boundary problem with finite Morse index in 3D.
method Proof of axial symmetry for finite index solutions.
result Finite index solutions to the Bernoulli problem in 3D are axially symmetric.
Paper constructs GCM spheres for Kerr family, removing symmetry restriction.
problem Establishing full nonlinear stability of Kerr family for perturbations.
method Introduction and construction of GCM hypersurfaces, removing symmetry restrictions.
result GCM spheres can be constructed for Kerr family without symmetry restrictions.
Sharp inequality proven for symmetric functions on a 4D sphere.
problem Proving a sharp Beckner's inequality for axially symmetric functions on S4. method Utilized pointwise properties of Gegenbauer polynomials.
result Sharp Beckner's inequality established for axially symmetric functions on S4. Axial-LOB predicts stock prices from LOB data using attention layers.
problem Predicting stock price from LOB data with long-range dependencies.
method Axial-LOB uses gated position-sensitive axial attention layers to incorporate global interactions.
result Axial-LOB achieves state-of-the-art performance in stock price prediction.
Smooth minimizers found for Willmore energy surfaces.
problem Finding minimizers for Willmore energy surfaces.
method Existence and smoothness established through axially symmetric surfaces with prescribed isoperimetric ratio.
result Existence and smoothness of minimizers proven.
In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter…
We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume…
For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian currents with the same boundary. This stands in sharp contrast with a results of Har…
3D Axial-Attention improves lung nodule classification accuracy.
problem Limited 3D attention in existing methods.
method Proposes 3D Axial-Attention network with 3D positional encoding.
result 3D Axial-Attention achieves state-of-the-art performance.
Defines axial curvatures for corank 1 singular manifolds in higher dimensions.
problem Characterizing singular n-manifolds in Rn+k with corank 1 singular points. method Using curvature locus and second fundamental form, defining up to l(n−1) axial curvatures. result Umbilic curvatures are absolute values of our axial curvatures.
We prove stability and exponential convergence of the Perfectly Matched Layer (PML) method for acoustic scattering on manifolds with axial analytic quasicylindrical ends. These manifolds model long-range geometric perturbations (e.g. bending or stretching) of tubular waveguides filled with homogeneous or inhomogeneous …
New minimal hypersurfaces found via transformations.
problem Finding new axially symmetric minimal hypersurfaces in 4D Minkowski space.
method Combining scaling symmetries and a non-obvious symmetry (analogous to Bianchi's transformation) to generate new hypersurfaces.
result Infinitely many axially symmetric minimal hypersurfaces can be generated from any given one.
We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in R3 with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additio…
Derives a Hamiltonian model for 3D axially symmetric magnetohydrodynamics.
problem Modeling of 3D axially symmetric magnetohydrodynamics.
method Hamiltonian formulation and matrix discretization.
result First discrete model for 3D magnetohydrodynamics compatible with underlying Lie-Poisson structure.
We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow an…
The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities phi: R^3Σ-> H^2_C, where Sigma is a subset of the axis of symmetry, and H^2_C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic m…
Proves positive mass theorem for specific initial data sets with corners.
problem Initial data sets with corners and non-smooth boundaries.
method Axially symmetric, maximal, complete initial data sets with two ends, proving for axially symmetric, simply connected, maximal, complete initial data sets with two ends.
result Proves positive mass theorem for specific initial data sets with angular momentum and charges.
Researchers define and evaluate quasi-local mass near axially symmetric null infinity.
problem Defining and evaluating quasi-local mass at null infinity.
method Using Bondi-van der Burg-Metzner coordinates, the researchers evaluate the Wang-Yau quasilocal mass on surfaces of unit size at null infinity of axi-symmetric spacetimes.
result Evaluation of quasi-local mass near axially symmetric null infinity.
Here are described the axiumbilic points that appear in generic one parameter families of surfaces immersed in R4. At these points the ellipse of curvature of the immersion, Little, Garcia - Sotomayor has equal axes. A review is made on the basic preliminaries on axial curvature lines and the associated axiumbilic poin…
Solves angles and multiplicities for metrics on spheres with singularities.
problem Angles and multiplicities of singularities in metrics on spheres.
method Analyzes Riemannian metrics with conic singularities and co-axial monodromy.
result Completes description of possible angles and multiplicities.
New theory shows how membranes can break symmetry.
problem Understanding symmetry breaking in membranes with boundaries.
method Applied bifurcation theory and reduced membrane equation.
result Existence of symmetry breaking bifurcation in membrane solutions.
We make a detailed study of the moduli space of winding number two (k=2) axially symmetric vortices (or equivalently, of co-axial composite of two fundamental vortices), occurring in U(2) gauge theory with two flavors in the Higgs phase, recently discussed by Hashimoto-Tong (hep-th/0506022) and Auzzi-Shifman-Yung (hep-…
We discuss the existence of Killing tensors for certain (physically motivated) stationary and axially symmetric vacuum space-times. We show nonexistence of a nontrivial Killing tensor for a Tomimatsu-Sato metric (up to valence 7), for a C-metric (up to valence 9) and for a Zipoy-Voorhees metric (up to valence 11). The …
We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when t…
Researchers create finite-time blow-up solutions for harmonic map flow into S2.
problem Constructing finite-time blow-up solutions for harmonic map flow into S2.
method Constructing finite time blow-up solutions with specific conditions and proving the blow-up behavior.
result First example of blow-up solution with a space-codimension 2 singular set.
Paper proves unique solutions for mean field equation on sphere.
problem Proving uniqueness of solutions for a specific equation on a sphere.
method Analyzing the mean field equation Δu=λ(1-e^u) on S^2, proving axially symmetry for even solutions.
result Zero is the only even solution for λ=6, implying rigidity of Hawking mass.
Smooth solution found for free boundary problem in higher dimensions.
problem One phase free boundary problem in R^N
method Constructing a smooth axially symmetric solution
result Existence of a smooth solution with catenoid type free boundary
Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.