By Federer and Fleming there exist at least one mass-minimizing normal current in every real-valued homology class of a Riemannian manifold. However the regularity of the mass-minimizing currents and their distributions may generally be quite complicated. In this paper we shall study how to construct nice metrics so th…
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25 results for “Mass-minimizing”
A theorem simplifies mass-minimizing flat chains' regularity.
problem Understanding the regularity of mass-minimizing flat chains.
method Simple condition for fundamental regularity principle.
result Fundamental regularity principle holds for mass-minimizing chains.
The paper extends local calibration pairs to global ones and finds mass-minimizing submanifolds.
problem Extending local calibration pairs to global ones in various Riemannian manifolds.
method Analyzing mass-minimizing properties and using conformal classes.
result Homologically mass-minimizing submanifolds exist in some Riemannian manifolds that cannot be calibrated by smooth calibrations.
New formula shows how causal vectors relate to mass-minimizing data.
problem Understanding mass-minimizing initial data sets and their geometry.
method Developed a new monotonicity formula for causal Killing vectors.
result Established strong maximum principles for the Lorentzian length.
Proves the validity of Bartnik's mass minimization conjecture for some specific 3-balls.
problem Bartnik's conjecture on mass minimization for asymptotically flat extensions of 3-balls.
method Analyzes Riemannian 3-balls with non-negative scalar curvature and proves the existence or non-existence of mass-minimizing extensions.
result Validates the second part of Bartnik's conjecture for some specific cases but disproves the first part.
New cones in 4D space found with minimal mass.
problem Finding mass-minimizing piecewise linear cones in 4D space.
method Mass minimization via Lipschitz maps, classification of candidates.
result No additional mass-minimizing cones found outside five known cases.
Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M. There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section s:M--> B is the mass of the image s(M) as a rectifiable n-current in B. Theorem…
The study examines vector fields with integer singularities in 3D balls.
problem Characterizing the strong -closure of vector fields with finitely many integer singularities.
method Characterization and decomposition of vector fields with finitely many integer singularities.
result Decomposition theorem for elements in , revealing information about mass-minimizing currents.
New insights into Bartnik mass from improvability of dominant energy scalar.
problem Characterizing Bartnik mass minimizing initial data sets.
method Introducing improvability concept, proving non-improvability consequences, and analyzing pp-wave counterexamples.
result Bartnik mass minimizing initial data sets are characterized, advancing conjectures.
Local minimality proven for stable free-boundary minimal hypersurfaces.
problem Proving local minimality for stable free-boundary minimal hypersurfaces.
method Using relative current setting and strict stability, proving local minimality among relative cycles.
result Local minimality of stable free-boundary minimal hypersurfaces in a small tubular neighborhood.
The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak-compactness theo…
In this article, we show that, for any compact 3-manifold, there is a 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…
Proves critical points of ADM mass correspond to specific initial data sets.
problem Finding initial data sets with fixed Bartnik boundary data.
method Proves existence of critical points on a Banach manifold.
result Critical points of ADM mass correspond to initial data sets with generalized Killing vector fields.
Study shows almost minimizing rectifiable chains in Hilbert space have regular points dense in their support.
problem Understanding the regularity of almost minimizing rectifiable chains in infinite dimensional spaces.
method Adapted Reifenberg's epiperimetric inequality and computations by Preiss to infinite dimensional space.
result The set of regular points is dense in the support of almost mass minimizing rectifiable chains.
Stationary polyhedral varifolds minimize area in two senses.
problem Minimizing area of polyhedral varifolds.
method Proves minimization of area through specific conditions.
result Stationary polyhedral varifolds minimize area in two senses.
Paper proves rigidity of 3-manifolds with boundary using modified Hawking mass.
problem Rigidity of 3-manifolds with boundary under specific geometric conditions.
method Area estimates for free boundary strictly stable two-disks, modified Hawking mass analysis.
result 3-manifolds with boundary are locally isometric to half anti-de Sitter-Schwarzschild manifold.
Maximizes capacity of extensions with fixed boundary data.
problem Maximizing the capacity of extensions with nonnegative scalar curvature.
method Using the method of Lagrange multipliers on the constraint space of scalar-flat extensions.
result Derives variational condition for maximal capacity extensions and proves they have constant scalar curvature.
The paper proves existence and partial regularity for Legendrian area-minimizing currents.
problem Existence and partial regularity of Legendrian area-minimizing currents.
method Local minimization and application to the Legendrian Plateau problem.
result Existence and partial regularity of solutions to the Legendrian Plateau problem.
The paper proves a generalized Stokes' Theorem for certain singular submanifolds.
problem Validity of Stokes' Theorem for singular submanifolds and differential forms.
method Combines Lebesgue integration with gauge integration techniques.
result Proves a generalized Stokes' Theorem for integral currents with finite Minkowski content.
New proof of Riemannian Penrose inequality in 3D without horizons.
problem Proving a new Riemannian Penrose inequality in 3D without horizons.
method Using the μ-bubble method to prove the inequality.
result The ADM mass is at least the square root of the area infimum of embedded surfaces.
Minimal networks minimize length and mass in certain configurations.
problem Finding minimal networks that minimize length and mass.
method Global and local calibrations to prove minimization properties.
result Minimal networks minimize mass and interfaces in partitions.
Derives evolution formula for quasi-local energy and proves rigidity theorem.
problem Rigidity of quasi-local mass in 3-manifolds with nonnegative scalar curvature.
method Evolution formula for Wang-Yau quasi-local energy, localized Penrose inequality.
result Rigidity theorem for compact 3-manifolds with nonnegative scalar curvature.
The paper extends semicontinuity of ADM mass to dimensions 2-7.
problem The semicontinuity of ADM mass in asymptotically flat metrics.
method Using recent work on the Riemannian Penrose inequality, the paper extends semicontinuity from dimension 3 to 7.
result The semicontinuity of ADM mass is proven for dimensions 2-7.
Study on stellar models' topology and mass using minimal surfaces.
problem Investigating the topology and mass of static stellar models.
method Analyzing stable free boundary minimal surfaces in static perfect fluid spaces.
result Proved non-existence of stable free boundary minimal surfaces and derived upper bounds for Hawking mass.
Calibrations help estimate volumes on odd spheres without gaps.
problem Estimating the minimum volume of tangent vector fields on odd spheres.
method Using a specific calibration and analyzing stable mass in the section class.
result No smooth unit field on has a graph that is -calibrated everywhere.