Study of large mass limits of G2 and Calabi-Yau monopoles on specific manifolds.
problem Understanding the behavior of monopoles in the large mass limit on G2 and Calabi-Yau manifolds.
method Developed a structure theory for the limit of SU(2) G2-monopoles and Calabi-Yau monopoles, extracting singular abelian G2-monopoles with Dirac singularities. result Proved an energy identity for monopole bubbles in the large mass limit.
Eigenvalues of Dirac operators on boundaries are large mass limits.
problem Eigenvalue calculation of Dirac operators on hypersurfaces.
method Limits of Euclidean Dirac operators with mass terms.
result Eigenvalues of boundary Dirac operators are limits of Euclidean ones.
In this paper, we will study the limiting behavior of the Brown-York mass of the coordinate spheres in an asymptotically flat manifold. Limiting behaviors of volumes of regions related to coordinate spheres are also obtained, including a discussion on the isoperimetric mass introduced by Huisken \cite{Huisken}. We will…
Study large mass G2 and Calabi--Yau monopoles, proving convergence and identifying key sets.
problem Large mass limits of G2 and Calabi--Yau monopoles on specific manifolds. method Common Θ-monopole framework, variational compactness theory, and finer analysis. result Identifies currents and shows saturation of calibration inequalities; defines sets S, Z, and C. Study on large mass monopoles focusing on their limiting behavior and properties.
problem Analyzing the limiting behavior of sequences of mSU(2) monopoles with large Yang--Mills--Higgs energies. method Bubbling analysis and finite cardinality bounds on the blow-up set and zero set.
result For large mass monopoles, the zero set and blow-up set coincide and are finite sets of points.
Mass in relativity linked to polyhedra geometry.
problem Understanding ADM mass in general relativity.
method Relating ADM mass to the total mean curvature and defect of dihedral angles of Riemannian polyhedra.
result Expressed n-dimensional mass as an integral of geometric quantities. Study mass and center of mass in flat 3-manifolds, proving existence of foliations.
problem Interplay between mass, center of mass, and isoperimetric quotients in asymptotically flat 3-manifolds.
method Adapted implicit function method and foliation techniques.
result Existence of foliations satisfying curvature conditions and unique relative isoperimetric surfaces.
The study examines the behavior of monopoles as their mass increases and finds that they abelianize near singular points.
problem The behavior of monopoles as their mass increases and the formation of singular points.
method Analysis of mass-renormalized energy measures and convergence of fields to a reducible monopole.
result The mass-renormalized energy measures concentrate at singular points, and the fields abelianize near these points.
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.
Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
Total mass equals limits of quasi-local mass integrals.
problem Calculating total mass on complex manifolds.
method Evaluated total mass via Ricci tensor limits of quasi-local mass integrals.
result Limits of quasi-local mass integrals equal total mass.
In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an as…
The article calculates the near horizon limit of Wang--Yau quasi-local mass.
problem Calculating the near horizon limit of quasi-local mass.
method Utilizing the properties of the mean curvature vector and optimal embedding equation.
result Existence and uniqueness of optimal embedding and continuity of quasi-local mass.
The paper examines mass aspects at future null infinity and limits of quasilocal mass.
problem Understanding mass aspects and limits of quasilocal mass at future null infinity.
method Review and extension of Bondi mass and mass loss formula in Bondi-Sachs coordinate system.
result New results about the limit of quasilocal mass of unit spheres at null infinity.
We study the limit of quasilocal mass defined in [4] and [5] for a family of spacelike 2-surfaces in spacetime. In particular, we show the limit coincides with the ADM mass at spatial infinity. The limit for coordinate spheres of a boosted slice of the Schwarzchild solution is computed explicitly and shown to give the …
Researchers calculate limits of angular momentum and center-of-mass at null infinity.
problem Evaluating angular momentum and center-of-mass limits for spacelike two-spheres approaching null infinity.
method Defined quasi-local angular momentum and center-of-mass by Chen-Wang-Yau, calculated limits for asymptotically flat spacetimes.
result Explicit expressions for angular momentum and center-of-mass at future null infinity derived in terms of spacetime observables.
This thesis discusses the Newtonian limit of General Relativity for static isolated systems with compactly supported matter. We call these systems "geometrostatic" to underline their geometric nature. We introduce new quasi-local notions of mass and center of mass that can be read off locally in the vicinity of the mat…
Researchers calculate quasi-local mass on unit spheres at infinity.
problem Computing quasi-local mass on unit spheres at spatial infinity.
method Developed new techniques to evaluate quasi-local mass.
result Leading order term of quasi-local mass recovers stress-energy tensor for vacuum spacetime.
Study shows mass distribution of random holomorphic sections follows a central limit theorem.
problem Understanding mass distribution of random holomorphic sections.
method Proved a central limit theorem for mass distribution of random holomorphic sections associated with positive line bundles.
result Almost every sequence of random holomorphic sections exhibits quantum ergodicity.
Lower semicontinuity of mass in 3D asymptotically flat manifolds proven.
problem Lower semicontinuity of mass in asymptotically flat 3-manifolds.
method Used Huisken's isoperimetric mass and modified weak mean curvature flow.
result Total mass is lower semicontinuous under C0 convergence. We show that the limit at infinity of the vector-valued Brown-York-type quasi-local mass along any coordinate exhaustion of an asymptotically hyperbolic 3-manifold satisfying the relevant energy condition on the scalar curvature has the conjectured causal character. Our proof uses spinors and relies on a Witten-type …
Deep learning predicts mass from images with sparse ground truth.
problem Accurately estimating mass from images with limited ground truth.
method Semi-supervised deep learning with gradient aggregation and sparse ground truth.
result Deep neural network accurately predicts mass from images.
In this paper, we will show that the limit of the Brown-York mass of a family of convex revolution surfaces in an asymptotically Schwarzschild manifold is the ADM mass.
Paper studies minimal hypersurfaces and their impact on compact manifolds with nonnegative scalar curvature.
problem Analyzing the boundary behavior of compact manifolds with nonnegative scalar curvature.
method Examines the effect of minimal hypersurfaces on the boundary of compact manifolds.
result Establishes an inequality relating mass, area of minimal hypersurfaces, and weighted total mean curvatures.
The paper shows how to create scalar flat metrics with very large ADM mass.
problem Understanding the ADM mass of scalar flat Kähler ALE spaces.
method Blowing up points in ALE spaces to increase ADM mass.
result It is possible to produce scalar flat metrics with arbitrarily large ADM mass.
New gravitational energy measure $\Q$ found in higher dimensions.
problem Characterizing local gravitational energy in higher dimensions.
method Study of quasilocal mass proposals in higher dimensions.
result New quantity $\Q$ replaces Bel-Robinson superenergy Q in vacuum limits. Paper proposes MMC to avoid high-density bias in clustering.
problem High-density bias in density-based clustering.
method Introduces mass distribution as a better foundation for clustering, proposing mass-maximization clustering (MMC).
result MMC avoids high-density bias and discovers clusters of arbitrary shapes, sizes, and densities.
The study examines mass drop and multiplicity in mean curvature flow.
problem Analyzing mass drop and multiplicity in mean curvature flow.
method Defined Brakke flow with variational inequality, proved mass drop conditions.
result Mass drop and multiplicity one conjecture are equivalent for Brakke flows.
The study proves the regularity of inverse mean curvature flow in specific geometric settings.
problem Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds.
method Utilizing the behavior of Hawking masses, the study shows star-shaped slices after a long time.
result The weak solution of inverse mean curvature flow becomes regular over time.
Mass in relativity linked to polyhedra geometry.
problem Mass in general relativity.
method Riemannian polyhedra geometry.
result Mass connected to polyhedra geometry.
A framework to quantify deployment risk in ML systems, especially for rare states.
problem Under-supported rare states in ML models lead to unreliable performance in unseen data.
method Blind-Spot Mass (B_n(tau)) using Good-Turing unseen-species estimation.
result Identifies and quantifies the risk of under-supported states in ML models.
The study proves stability of the positive mass theorem for Kähler manifolds.
problem Stability of the positive mass theorem for Kähler manifolds.
method Integral inequality and stability results for ADM mass on AE Kähler manifolds.
result Stability of the positive mass theorem for Kähler manifolds under certain conditions.
Computes quasi-local mass at null infinity using Bondi-Sachs coordinates.
problem Global properties of quasi-local mass at null infinity.
method Evaluation of Wang-Yau quasi-local mass on unit spheres in Bondi-Sachs coordinates.
result Quasi-local mass is related to the news function in Bondi-Sachs coordinates.
We construct the first nontrivial examples of Calabi-Yau monopoles. Our main interest on these, comes from Donaldson and Segal's suggestion \cite{Donaldson2009} that it may be possible to define an invariant of certain noncompact Calabi-Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metri…
Researchers compute limits of Kähler-Einstein forms on degenerating manifolds.
problem Understanding limits of Kähler-Einstein forms on degenerating manifolds.
method Hybrid convergence of Kähler-Einstein measures using algebro-geometric limits.
result Limit measure is a weighted sum of Dirac masses at divisorial valuations.
Study shows spherical hyperbolic manifolds almost rigidly converge to hyperbolic space.
problem Almost rigidity of positive mass theorem for spherical hyperbolic manifolds.
method Intrinsic flat distance to prove convergence.
result Spherically symmetric asymptotically hyperbolic manifolds converge to hyperbolic space if mass limit is zero.
Study shows a mass quantity for C0 metrics that agrees with ADM mass.
problem Understanding ADM mass for C0 metrics and its behavior under Ricci-DeTurck flow. method Developed a C0 mass quantity and analyzed its behavior under Ricci-DeTurck flow. result The C0 mass at infinity is independent of coordinate charts and has controlled distortion under Ricci-DeTurck flow. SPT predicts age and mass of red giants from spectra.
problem Challenges in age and mass estimation of red giants using traditional methods.
method SPT framework with Multi-head Hadamard Self-Attention and Mahalanobis distance-based loss function.
result Remarkable age and mass estimations with low errors and uncertainties.
In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previo…
Continuous metrics on R^3 with specific properties have non-negative harmonic mass.
problem Proving non-negativity of mass for continuous metrics.
method Defining harmonic mass and using properties of approximating smooth metrics.
result The harmonic mass of continuous metrics is non-negative.
In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some …
In this paper we characterize the intrinsic geometry of apparent horizons (outermost marginally outer trapped surfaces) in asymptotically flat spacetimes; that is, the Riemannian metrics on the two sphere which can arise. Furthermore we determine the minimal ADM mass of a spacetime containing such an apparent horizon. …
Upper bounds on Bartnik mass for non-negatively curved spheres.
problem Bounding Bartnik mass for non-negatively curved spheres.
method Establishing upper bounds using non-negative Gauss curvature.
result Upper bounds on Bartnik mass approach Hawking mass under certain conditions.
Study the mass of flat 3-manifolds with boundary using specific methods.
problem Calculate the mass of asymptotically flat 3-manifolds with boundary.
method Use the method of Bray-Kazaras-Khuri-Stern to derive a mass formula.
result Derive sufficient conditions for the positivity of the mass.
The paper proves Zimmer's conjecture for non-uniform lattices by controlling mass escape and Lyapunov exponents.
problem Proving Zimmer's conjecture for non-uniform lattices in higher-rank semisimple Lie groups.
method Establishes finiteness of low-dimensional actions, introduces novel techniques to control mass escape and Lyapunov exponents.
result Proves Zimmer's conjecture for many non-uniform lattices, improving previous results.
Study mass transport in low-diffusivity using Lagrangian coordinates.
problem Mass preserving transport of passive tracers in low-diffusivity limit.
method Lagrangian coordinates, time-averaged diffusion equation, weighted manifold structure.
result Leading order asymptotics extend to dominant nontrivial singular value in low-diffusivity limit.
Given a spacelike 2-surface Σ in a spacetime N and a constant future timelike unit vector T0 in R3,1, we derive upper and lower estimates of Wang-Yau quasilocal energy E(Σ,X,T0) for a given isometric embedding X of Σ into a flat 3-slice in R3,1. The quantity E(Σ,X,T0) itself depends …
We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivit…