Proves properties of full mass currents in big cohomology classes.
problem Characterizing and understanding full mass currents in big cohomology classes.
method Uses a characterization of full mass currents in terms of the envelope of their singularity type and develops the theory of weak geodesics in big cohomology classes.
result Shows the inclusion of certain sets of functions and currents and characterizes big classes with additive full mass currents.
Maps preserving mass and injective on boundary are isometries.
problem Stability of mass-preserving maps in integral current spaces.
method Proving rigidity of mass-preserving 1-Lipschitz maps.
result Maps preserving mass and injective on boundary are isometries.
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…
Study on fractional mass for codimension-two currents, proving equi-coercivity and Γ-convergence.
problem Defining and studying fractional mass for codimension-two currents on manifolds.
method Energy minimization with Jacobian constraint, equi-coercivity, Γ-convergence, weak linking.
result Equivalence of two formulations of fractional mass, improved regularity for s-harmonic maps. Integral currents with boundary of finite mass are integral.
problem Integral currents with boundary of finite mass are integral.
method De Giorgi's structure theorem for integer-valued BV functions and a cylindrical projection argument. result Integral currents with boundary of finite mass are integral.
Stability of positive mass theorem for hyperbolic graphs proven.
problem Proving stability of positive mass theorem for asymptotically hyperbolic graphs.
method Adapting ideas from previous work on asymptotically flat graphs to hyperbolic graphs.
result Stability of positive mass theorem for a class of n-dimensional asymptotically hyperbolic graphs.
Stability results for complex Monge-Ampère equations in various classes.
problem Stability of solutions to complex Monge-Ampère equations.
method Weak stability results followed by Ck,α stability proofs. result Proves stability of solutions in relative full mass classes and on quasi-projective varieties.
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. The paper solves a partial Plateau problem using H-mass.
problem Finding a surface of least area with a partially specified boundary.
method Minimizing H-mass over scans with boundary. result Existence of a rectifiable minimizer for the H-mass problem. 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.
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…
Solves a problem posed by Brezis and Mironescu about least mass of area-minimizing currents.
problem Least mass of area-minimizing currents with a given boundary.
method Demonstrates the value of the least mass and compares it to the infimum of areas of smoothly immersed submanifolds.
result The least mass of area-minimizing currents equals the infimum of areas of smoothly immersed submanifolds with the same boundary.
The Bartnik mass increases as spacetime evolves.
problem Understanding the evolution of the Bartnik mass in spacetime.
method Computing the derivative of the Bartnik mass along evolving surfaces under the assumption of the dominant energy condition.
result The Bartnik mass of evolving surfaces is monotone non-decreasing.
Deep neural network predicts black hole merger remnants with high accuracy.
problem Estimating the properties of black hole merger remnants.
method Trained on a dataset of binary black hole simulations, a deep neural network predicts mass and spin with high precision.
result The network predicts remnant black hole mass and spin with errors less than 0.04% and 0.3% respectively, and reduces errors in precessing cases to half.
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.
Identifies conditions for multiple invariant probabilities in Markov kernels.
problem Global irreducibility and recurrence do not guarantee uniqueness of invariant probabilities.
method Uses Jordan decomposition of the difference of two invariant probabilities.
result A Markov kernel has more than one invariant probability if and only if it admits a visible absorbing decomposition.
We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we…
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. …
We extend Brill's positive mass theorem to a large class of asymptotically flat, maximal, U(1)2-invariant initial data sets on simply connected four dimensional manifolds Σ. Moreover, we extend the local mass angular momenta inequality result Ref [1] for U(1)2 invariant black holes to the case with nonzero stre…
Recently, a new embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric spaces. To this end we introduce another variant of the Ambrosio--Kirchheim theo…
We adapt the theory of currents in metric spaces, as developed by the first-mentioned author in collaboration with B. Kirchheim, to currents with coefficients in Z_p. Building on S. Wenger's work in the orientable case, we obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these inequalities to prov…
The Riemannian hemisphere has a lower bound for its mass.
problem Estimating the mass of surfaces spanning a circle.
method Constructing a differential form with a stationary comass norm on the hemisphere.
result The mass of surfaces spanning a circle has a lower bound of 2π plus a second-order term. The study examines vector fields with integer singularities in 3D balls.
problem Characterizing the strong Lp-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 LZ1(B), revealing information about mass-minimizing currents. Solves a complex Monge-Ampère equation on compact Hermitian manifolds.
problem Solving a specific Monge-Ampère equation on compact Hermitian manifolds.
method Uses complex Monge-Ampère equation and fixed potential approach.
result Shows the existence and uniqueness of a solution in a specific class.
Given a transportation cost c:M×Mˉ→R, optimal maps minimize the total cost of moving masses from M to Mˉ. We find a pseudo-metric and a calibration form on M×Mˉ such that the graph of an optimal map is a calibrated maximal submanifold. We define the mass of space-like current…
Unique tangent cones found for boundary points of 2D almost-minimizing currents.
problem Characterizing boundary points of two-dimensional almost-minimizing currents.
method Combining epiperimetric inequality and almost-monotonicity formula.
result Tangent cones at singular boundary points are unique.
New methods using spacetime harmonic functions solve geometric inequalities.
problem Geometric inequalities involving mass in spacetime.
method Utilizing spacetime harmonic functions and other elliptic equations.
result Novel concept of total mass and proof of positive mass theorem.
Researchers describe the mass of conformal differential operators in terms of their asymptotic expansions.
problem Understanding the mass of conformal differential operators and its invariance under conformal transformations.
method Explicit description of the full asymptotic expansion of the Schwartz kernel of complex powers of m-Laplace type operators. result The mass of conformal differential operators is a conformal invariant in odd dimensions when the kernel is trivial.
Bayesian model predicts drip-line locations in heavy calcium isotopes.
problem Determining the neutron drip line in the Ca region of heavy nuclei.
method Bayesian model averaging with Gaussian-process-based extrapolations.
result Predicted posterior probabilities for drip-line locations in heavy calcium isotopes.
We prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current; we derive explicit formulae for the boundary, the mass and the first variations a…
We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an exp…
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…
We study a simple modification to the conventional time of flight mass spectrometry (TOFMS) where a \emph{variable} and (pseudo)-\emph{random} pulsing rate is used which allows for traces from different pulses to overlap. This modification requires little alteration to the currently employed hardware. However, it requi…
Analyzes Kähler-Einstein metrics on families of Fano varieties.
problem Establishing Kähler-Einstein metrics on Fano varieties in families.
method Analytic method to show unique Kähler-Einstein metrics on neighboring fibers.
result Uniform a priori estimates and continuous variation of Kähler-Einstein potentials.
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.
The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be realized as graphical hypersurfaces in Rn+1. Specifically, for an asympto…
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.
Proposes a new quasi-local mass for timelike 2-surfaces in spacetimes.
problem Need a mass definition for 2-surfaces with timelike mean curvature.
method Adopts Wang-Yau's quasi-local energy framework, modifies for timelike mean curvature.
result Yields a positive definite surface energy density and divergence-free current.
Characterizes infinite harmonic maps using 1-currents.
problem Defines critical points of a non-differentiable functional.
method Uses subdifferential and geometric condition in terms of 1-currents.
result Geometric condition equivalent to criticality in terms of 1-currents.
In the first half of this article, we survey the new quasi-local and total angular momentum and center of mass defined in [9] and summarize the important properties of these definitions. To compute these conserved quantities involves solving a nonlinear PDE system (the optimal isometric embedding equation), which is ra…
Study distances between special functions on Kähler manifolds.
problem Measuring distances between plurisubharmonic functions on Kähler manifolds.
method Introduce a distance function ρ[u,v] and explore its properties.
result Properties of ρ[u,v] generalize Darvas's metrics.
Bayesian methods improve nuclear mass predictions for unstable nuclei.
problem Improving predictions of nuclear masses far from stability.
method Bayesian Gaussian processes and neural networks applied to 10 models.
result Significant reduction in rms deviation from experimental data.
The Gilbert-Steiner problem is a mass transportation problem, where the cost of the transportation depends on the network used to move the mass and it is proportional to a certain power of the "flow". In this paper, we introduce a new formulation of the problem, which turns it into the minimization of a convex function…
Study shows Lelong numbers vanish for certain currents in weakly hyperbolic foliations.
problem Analyzing Lelong numbers for currents in weakly hyperbolic foliations.
method Local and global analysis of directed positive harmonic currents and currents directed by foliations.
result Lelong numbers of currents at the singularity vanish.
Study on sphere-valued maps, proving energy convergence and current limits.
problem Understanding the behavior of sphere-valued Sobolev maps as their energy grows.
method Proving Gamma-convergence of p-energies to the mass of an integral current. result Jacobian convergence to an area-minimizing current in a cobordism class.
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.
We define generalized currents associated with immersions of abstract oriented solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geo…
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.