The paper proves conditions for isoperimetric regions in curved spaces.
problem Finding isoperimetric regions in curved spaces with specific curvature and growth conditions.
method Combining asymptotic mass decomposition, sharp isoperimetric inequality, and concavity property.
result Isoperimetric regions always exist under certain conditions.
New criteria detect anomaly detection algorithms without labeled data.
problem Lack of labeled data for evaluating anomaly detection algorithms.
method Developed two new criteria based on Excess-Mass and Mass-Volume curves, and a feature sub-sampling methodology.
result Empirically validated new criteria outperform classical ROC and PR curves in non-labeled data scenarios.
The paper introduces a new method to rank anomalies in multivariate data.
problem Ranking multivariate unlabeled observations based on their degree of abnormality.
method Formulates the problem as a M-estimation problem using the Mass Volume curve (MV curve).
result Established generalization bounds for the MV curve estimation.
Defines a new geometric quantity for hyperbolic manifolds, showing it's well-defined and invariant.
problem Defining a mass for asymptotically hyperbolic manifolds under weaker conditions.
method Volume-renormalized mass defined as a linear combination of ADM mass and renormalized volume.
result Volume-renormalized mass is well-defined and diffeomorphism invariant under weaker conditions.
Defines circumcenter of mass for polytopes without triangulation.
problem Defining circumcenter of mass for polytopes without triangulation.
method Investigates how volumes of polytopes change under Möbius transformations.
result Provides a definition of circumcenter of mass independent of triangulation.
New method calculates volume-renormalized mass from Hamiltonian perspective.
problem Calculating volume-renormalized mass for asymptotically hyperbolic manifolds.
method Using Michel's mass invariants and a reduced Hamiltonian perspective, the volume-renormalized mass is deduced.
result The reduced Hamiltonian recovers the volume-renormalized mass and its variations.
The paper calculates mass and volume of Einstein metrics in four dimensions.
problem Calculating mass and volume of Einstein metrics in four dimensions.
method Using Green's function and conformal laplacian, the paper expresses ADM mass as an integral and proves a mass-volume inequality.
result Proves a lower bound for the mass of a metric in terms of its volume, and various mass gap theorems.
New metrics found in hyperbolic manifolds as volume-minimizers.
problem Finding critical points of volume-renormalized mass.
method Critical points of the volume-renormalized mass over asymptotically hyperbolic manifolds.
result V-static metrics are critical points of volume-renormalized mass.
New definition of metric current yields Finsler geometry volume densities.
problem Defining volume functionals from Finsler geometry.
method Proposed a new definition of metric current and showed its utility.
result Obtained a family of extendibly convex volume densities.
Generalizes inequality for complete manifolds involving homology classes.
problem Volume and simplicial volume inequality for closed manifolds.
method Extends inequality to ℓ1-norm of homology classes on complete manifolds. result Inequality involving critical exponent and mass of homology classes.
New mass inequalities and proofs for causal variational principles.
problem Proving new mass inequalities for causal variational principles.
method Proved a new inequality for minimizers of causal variational principles and applied it to prove the positive mass theorem.
result Introduced a positive quasilocal mass and proved new mass inequalities.
Study proves inequalities for mass-capacity on curved spaces.
problem Proving nonnegativity and positive lower bounds of mass on curved spaces.
method Applying mass-capacity inequalities from \cite{M22} to manifolds with nonnegative scalar curvature.
result Sufficient conditions for nonnegativity and positive lower bounds of mass.
New mass definition linked to ADM mass for general metrics.
problem Defining mass for metrics with low regularity.
method Using isocapacitary inequality to define total mass.
result Inequality between new mass and ADM mass proved.
The paper proves a new inequality linking mass and volume in 3D space.
problem Relating mass and volume in 3D space with a sharp inequality.
method Using a monotonicity formula for level sets of a 3-harmonic function.
result Sharp lower bound for ADM mass in terms of Euclidean volume of Ω.
Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.
problem Understanding conformal metrics with finite total Q-curvature.
method Introduces conformal mass and provides necessary and sufficient conditions for normality.
result Derives volume comparison theorems and proves a positive mass type theorem related to Q-curvature.
Study volumes of Bott-Chern classes on complex manifolds.
problem Understanding volumes of transcendental Bott-Chern classes.
method Extending non-pluripolar products to quasi-positive currents, establishing quasi-monotonicity of Monge-Ampère masses, and solving degenerate complex Monge-Ampère equations.
result Positive answer to Demailly-Păun-Boucksom conjecture regarding bounded mass property.
Proves positive mass theorems for specific types of curved spaces.
problem Analyzing mass in curved spaces with boundaries.
method Proves positive mass theorems for specific types of curved spaces with boundaries.
result Establishes conditions under which mass is positive in these spaces.
Paper proves flat 3-manifolds with positive mass have unique isoperimetric surfaces.
problem Finding unique isoperimetric surfaces in flat 3-manifolds.
method Used 'fill-in' argument and sharp isoperimetric inequality.
result Each leaf of the canonical foliation is the unique isoperimetric surface.
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 connects isoperimetric sets, mass concepts, and nonnegative scalar curvature.
problem Understanding mass concepts in nonnegative scalar curvature manifolds.
method Review and analysis of isoperimetric sets and their role in mass concepts.
result Equivalence among mass concepts holds for isoperimetric sets with connected boundaries.
Paper proves rigidity of CMC surfaces in curved 3-manifolds.
problem Rigidity of CMC surfaces in positive curved 3-manifolds.
method Assumptions of surface being approximately round or invariant under even symmetry, and use of Hawking mass.
result Rigidity results for stable CMC surfaces with zero Hawking mass.
In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold M^n is bounded from above by (n+2)!d/3,…
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.
New isoperimetric surfaces found in flat 3-manifolds.
problem Finding unique isoperimetric surfaces in flat 3-manifolds.
method Analyzing scalar curvature and mass properties.
result Each leaf of the foliation is uniquely isoperimetric.
The Planck mass can be derived from gravitational potential behavior in compactifications.
problem Deriving the Planck mass from gravitational potential behavior in compactifications.
method Physical considerations and Weyl law application to gravitational potential behavior.
result The Planck mass can be reconstructed from the asymptotics of the masses of spin 2 Kaluza--Klein modes.
Characterizes curves for minimal surfaces in de Sitter space.
problem Minimal surfaces in de Sitter space.
method Variational problem to find critical points of center of mass.
result Curves are critical points of center of mass.
Study semicontinuity of capacity in non-smooth spaces using intrinsic flat convergence.
problem Investigate semicontinuity of capacity in non-smooth spaces.
method Analyze sequences of local integral current spaces converging in the pointed Sormani-Wenger intrinsic flat sense.
result Prove upper semicontinuity of capacity for balls and Lipschitz sublevel sets under volume-preserving convergence.
Article connects Besse's conjecture to positive mass theorem and Brown-York mass.
problem Critical point of Hilbert-Einstein functional with constraints.
method Positive mass theorem and Brown-York mass analysis.
result Connection between Besse's conjecture and positive mass theorem.
New positive mass theorem for hyperbolic 3-manifolds using Green functions.
problem Proving a positive mass theorem for asymptotically hyperbolic 3-manifolds.
method Using a monotonicity formula for the Green function of the Laplace operator.
result Established a new positive mass theorem for three-dimensional manifolds.
The paper proves stability of manifolds with boundary under volume and distance constraints.
problem Stability of manifolds with boundary under volume and distance constraints.
method Volume preserving intrinsic flat convergence of metrics with boundary constraints.
result The stability of manifolds with boundary under volume and distance constraints is proven.
The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.
problem Stability and instability of Poincaré-Einstein metrics.
method Variant of expander entropy for asymptotically hyperbolic manifolds, local positive mass theorem, volume comparison.
result Characterization of stability and instability in terms of local positive mass theorem and volume comparison.
The paper studies entropy and mass loss in geodesic flows on curved spaces.
problem Entropy and mass loss in geodesic flows on negatively curved manifolds.
method Ergodic theory, critical exponents of parabolic subgroups, pressure of potentials.
result Entropy is upper semicontinuous with no mass loss, but fails with mass loss due to critical exponents.
Stability theorem for axisymmetric manifolds with nonnegative scalar curvature.
problem Stability of positive mass theorem for specific types of manifolds.
method Proved stability in W1,p sense with technical assumptions. result Derived estimates for volumes, areas, and distances.
Improved clustering and anomaly detection using latent space conditioning.
problem Anomaly detection on unlabeled data.
method Conditional latent space variational autoencoder (cLSVAE) that separates latent space by conditioning on data labels.
result The method outperforms typical variational autoencoders in clustering and anomaly detection.
Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.
problem Proving the uniqueness of the mass center system in non-Euclidean geometries and deriving a generalized Pappus' theorem.
method Revisiting and simplifying G.A. Galperin's proof, extending the mass center system to manifolds, and deriving a generalized Pappus' theorem.
result Unified and simpler proofs for Pappus' theorem in Euclidean, spherical, and hyperbolic geometries.
Study shows trace and mass of subcritical GJMS operators on manifolds.
problem Characterize the trace and mass of subcritical GJMS operators.
method Analyzes zeta-regularized trace of inverse GJMS operator, uses positive mass theorems.
result Supremum of trace is greater than or equal to standard sphere, attains constant mass under certain conditions.
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…
Deep learning ν-net automates cardiac MRI segmentation for accurate mass and function parameters.
problem Challenging image segmentation of biventricular cardiac anatomy.
method Deep neural network training on 253 manually segmented cases, evaluated on 1000 cases.
result State-of-the-art performance in LV and RV EF, VM measurements.
Formula calculates mass using cube faces and edges.
problem Measuring mass of 3-manifolds.
method Cube faces and edges, mean curvature, dihedral angle, geodesic curvature, angle defect.
result Mass formula connects to Gromov's theory and Gauss-Bonnet theorem.
The classification of solutions of the static vacuum Einstein equations, on a given closed manifold or an asymptotically flat one, is a long-standing and much-studied problem. Solutions are characterized by a complete Riemannian n-manifold (M,g) and a positive function N, called the lapse. We study this problem o…
Computes the outer mass of small metric spheres in time-symmetric slices.
problem Computing the outer mass of small metric spheres in time-symmetric slices.
method Analyzes the deviation from a standard sphere, estimates mass of a static vacuum extension, and uses geodesic ball shrinking to a point as an application.
result Outer mass to first order in the data's deviation from the standard sphere, with an upper bound obtained by estimating the mass of a static vacuum extension.
In this article, we show that, for any compact 3-manifold, there is a C1 volume-minimizing one-dimensional foliation. More generally, we show the existence of mass-minimizing rectifiable sections of sphere bundles without isolated "pole points" in the base manifold. This same analysis is used to show that the exam…
Lower semicontinuity of ADM mass proven for a weaker convergence type.
problem Behavior of ADM mass under weak convergence types.
method Intrinsic flat convergence, smooth manifolds converging to local integral current spaces, Huisken's isoperimetric mass.
result Lower semicontinuity of ADM mass proven for F convergence. Rigidity results for Hawking mass in curved spaces with bounds on Bartnik capacity.
problem Rigidity of surfaces in curved spaces with mass bounds.
method Analyzing Hawking mass and applying rigidity results to specific geometric settings.
result Explicit lower bounds on Hawking and Bartnik masses in non-flat spaces.
Let (M,g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K of M, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid K. This complements the work of G. Huisken and S.-T. Yau and J. …
The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.
problem Proving positivity of simplicial volume for specific types of 4-manifolds.
method Using the Gauss-Bonnet theorem for Riemannian simplices, the paper shows that for closed nonpositively curved 4-manifolds with nonzero Euler characteristic, simplicial volume is positive.
result The paper confirms conjectures about the relationship between simplicial volume and Euler characteristic for four-dimensional manifolds.
New proof shows nonpositive curved manifolds have positive simplicial volume.
problem Proving positive simplicial volume for nonpositive curved manifolds.
method Using negative $([ rac{n}{4}]+1)$-Ricci curvature condition.
result Closed manifolds with nonpositive curvature have positive simplicial volume.
The study of equal-volume polygons in 3D space and their affine invariants.
problem Estimating projective invariants of planar curves.
method Developing a theory of discrete affine invariants from equal-volume polygons.
result Equal-volume polygons can be used to estimate projective invariants of a planar curve.