Proves intrinsic geometry matches for Finsler structures.
problem Matching intrinsic geometry with differential structures for Finsler structures.
method Proves equivalence of intrinsic distance and differential structures for weak upper semicontinuous admissible Finsler structures.
result Intrinsic geometry and differential structures coincide for Finsler structures.
We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a …
Study proves upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.
problem Proving upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.
method Analyzing a weighted eigenvalue problem and using a Lorentz-Sobolev inequality to study eigenfunctions and index/nullity in neck regions.
result Upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces proved.
Proves existence and regularity of spherical minimizers for lipid membrane energy.
problem Existence and regularity of minimizers for the Canham-Helfrich energy.
method Establishes lower semicontinuity and proves existence through weak convergence of immersions.
result Proves existence and regularity of minimizers for the Canham-Helfrich energy on spheres.
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. Recall that Federer-Fleming defined the notion of flat convergence of submanifolds of Euclidean space to solve the Plateau problem. Here we prove the upper semicontinuity of Neumann eigenvalues of the submanifolds when they converge in the flat sense without losing volume. With an additional condition on the boundaries…
Study energy functional's second variation on manifolds, proving Morse index bounds.
problem Understanding the stability and index of phase transition interfaces.
method Extending gradient theory to minimal hypersurfaces, proving upper semicontinuity of stability operator eigenvalues.
result Upper bounds for the Morse index of limit interfaces without multiplicity or orientability conditions.
The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger--Gromov convergence (without any symmetr…
Study how nodal domains change on surfaces under perturbations.
problem How eigenfunction nodal domains change on surfaces under smooth perturbations.
method Sector/graph count near nodal critical points, upper semicontinuity proof, branch-free on spectral clusters, wavelength-scale analysis.
result Upper semicontinuity of nodal domain count, no new domains created at wavelength scale, stable count in noncritical cases.
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.
New metric measure space theory for Lipschitz constants.
problem Defining and characterizing Cheeger energy in metric measure spaces.
method Adapting Cheeger theory to intrinsically Lipschitz sections.
result Characterization of intrinsic Cheeger energy in terms of relaxed slope.
We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. …
We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a byproduct, we get that any circle decomposition of a surface is upper semicontinuou…
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.
Study market models without concavity assumptions, proving representability and extending no-arbitrage concepts.
problem Market models without concavity assumptions.
method Defining market models, proving representability, extending no-arbitrage concepts.
result Market models can be represented as normal integrands under sequential upper-semicontinuity.
Study finds minimizers for complex membrane models without symmetry assumptions.
problem Minimizing the Canham-Helfrich functional in multiple phases for heterogeneous biological membranes.
method Reformulated as oriented curvature varifolds, proving existence without symmetry assumptions.
result Existence of minimizers for single- and multiphase models under constraints.
We show that if G is an upper semicontinuous decomposition of Rn, n≥4, into convex sets, then the quotient space Rn/G is a codimension one manifold factor. In particular, we show that Rn/G has the disjoint arc-disk property.
Investigates stability properties of Haezendonck-Goovaerts premium principles in Orlicz spaces.
problem Stability properties of Haezendonck-Goovaerts premium principles in various Orlicz spaces.
method Analysis of stability properties including Fatou and Lebesgue properties, and continuity with respect to Φ-weak convergence. result Haezendonck-Goovaerts principles satisfy the Fatou property and Lebesgue property under certain conditions.
Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.
problem Positive entropy actions by higher-rank lattices in Lie groups.
method Analysis of sub-actions, fiber entropy upper semicontinuity, and conjugacy arguments.
result Actions by higher-rank lattices in SL(n,R) are conjugate to affine actions on (infra-)tori. The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this…
We introduce a setup of model uncertainty in discrete time. In this setup we derive dual expressions for the super--replication prices of game options with upper semicontinuous payoffs. We show that the super--replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the c…
In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is…
The article proves a lower semicontinuity property of holonomy maps.
problem Continuity properties of holonomy maps on manifolds.
method Examined continuity properties of holonomy class and restricted holonomy class maps.
result The restricted holonomy class map is lower semicontinuous.
A semicontinuous semifinite trace is constructed on the C*-algebra generated by the finite propagation operators acting on the L^2-sections of a hermitian vector bundle on an amenable open manifold of bounded geometry. This trace is the semicontinuous regularization of a functional already considered by J. Roe. As an a…
Study Galois groupoids of vector fields, proving lower semicontinuity.
problem Computing Galois groupoids for general parameter values of Painlevé equations.
method Prove lower semicontinuity of Galois groupoids of vector fields.
result Results can compute Galois groupoids for general parameter values of Painlevé equations.
New method for practical hedging under uncertainty in continuous time models.
problem High minimal superhedging price for practical use in continuous-time models.
method Relaxed hedging criterion based on acceptable shortfall risks, combining aggregation and convex dual representation theorems.
result Derivation of duality results for minimal price on discounted claims.
Paper introduces a new time separation function for C0 spacetimes.
problem Lower semicontinuity of time separation function for C0 spacetimes. method Introduced nearly timelike curves to ensure lower semicontinuity.
result Lower semicontinuous time separation function for C0 spacetimes. The normalized volume is lower semicontinuous in klt singularities.
problem Lower semicontinuity of normalized volumes in klt singularities.
method Flat family of klt singularities, alternative characterization of K-semistability.
result K-semistability is very generic or empty in log Fano pairs.
The first result is the semicontinuity of automorphism groups for the collection of complex two-dimensional bounded pseudoconvex domains with smooth boundary of finite D'Angelo type. The method of proof is new so that it simplifies the previous proof of earlier semicontinuity theorems on bounded strongly pseudoconvex d…
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.
We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measu…
Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtaine…
New method for superhedging without assuming continuous claims.
problem Superhedging without assuming upper semicontinuous contingent claims.
method Established a generalized duality for model-free superhedging using Choquet's capacitability theorem.
result Generalized duality for superhedging given marginal distributions without continuity assumptions.
The L^p norm of Poisson brackets is lower semicontinuous on surfaces for p < ∞.
problem Lower semicontinuity of L^p norms of Poisson brackets on surfaces.
method Proof of lower semicontinuity for Cc∞(M) functions on surfaces with dimM=2 and p<∞. result The functional (F,G)↦∥{F,G}∥Lp(M) is lower semicontinuous with respect to the C0-norm on Cc∞(M) when dimM=2 and p<∞. New method decomposes submartingale systems for BSDEs with weak constraints.
problem Tackles decomposition of submartingale systems for BSDEs with weak constraints.
method Introduces Yg,ξ-submartingale systems and proves a Mertens decomposition using an original approach. result Proves a Mertens decomposition for Yg,ξ-submartingale systems. We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in Rn. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity…
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. We use purely topological methods to prove the semicontinuity of the mod 2 spectrum of local isolated hypersurface singularities in Cn+1, using Seifert forms of high-dimensional non-spherical links, the Levine--Tristram signatures and the generalized Murasugi--Kawauchi inequality obtained in earlier work …
To any utility maximization problem under transaction costs one can assign a frictionless model with a price process S∗, lying in the bid/ask price interval [S,Sˉ]. Such process S∗ is called a \emph{shadow price} if it provides the same optimal utility value as in the original model with bid-as…
Sharp upper bounds found for solutions of a specific equation on Riemannian manifolds.
problem Finding upper bounds for solutions of a specific equation on Riemannian manifolds.
method Proved sharp upper estimates of weak subsolutions to the Leibenson equation on Riemannian manifolds with non-negative Ricci curvature.
result Improved and proved a conjecture about upper bounds for solutions of the Leibenson equation.
We use topological methods to study various semicontinuity properties of spectra of singular points of plane algebraic curves and of polynomials in two variables at infinity. Using Seifert forms and the Tristram--Levine signatures of links, we reprove (in a slightly weaker version) a result obtained by Steenbrink and V…
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
Study proves upper bounds for solutions on Riemannian manifolds.
problem Proving upper bounds for solutions of Leibenson's equation on Riemannian manifolds.
method Proved upper bounds equivalent to a euclidean-type Sobolev inequality.
result Upper bounds for solutions of Leibenson's equation on Riemannian manifolds are equivalent to euclidean-type Sobolev inequalities.
The study explores the strengths and weaknesses of models that generalize from weak to strong supervision.
problem Understanding the limitations and capabilities of models that generalize from weak to strong supervision.
method Theoretical analysis and experimental validation in both classification and regression settings.
result Theoretical bounds reveal the importance of strong generalization and calibration of the weak model and a careful balance in the training process.
Excises interesting subsets from symplectic manifolds.
problem Excision of interesting closed subsets from symplectic manifolds.
method Time-independent incomplete Hamiltonian flows.
result Generalizes a result about excision of a ray.
Extends risk measure theory to general Orlicz spaces.
problem Applying risk measure theory to non-standard spaces.
method Generalizes results from bounded random variables to general Orlicz spaces, proving new characterizations and extensions.
result Characterizations and extensions of the Fatou property and Kusuoka representation in Orlicz spaces.
The paper introduces generalized Lelong numbers for currents and their applications in intersection theory.
problem Defining and studying generalized Lelong numbers for currents in intersection theory.
method Formulating generalized Lelong numbers for closed smooth (j,j)-forms, defining horizontal dimension, and establishing properties and formulas.
result Effective sufficient conditions for defining and continuity of intersections of positive closed currents.
New metric defines surface shapes, minimizing area and angle distortions.
problem Defining and measuring the shape of high genus surfaces.
method Defined a metric space, introduced energies for area and angle distortions, showed minimizers by lower semicontinuity.
result Energy minimizers in surface shape space correspond to quasiconformal homeomorphisms.