Solves the asymptotic Plateau problem in hyperbolic space for specific curvature.
problem Existence of complete hypersurfaces with prescribed asymptotic boundary.
method Curvature estimates.
result Solves the problem for a wider range of curvature values.
Geodesic lines with specific boundaries found on a special type of manifold.
problem Existence of geodesic lines with prescribed asymptotic boundaries.
method Proper exponential map assumption, solution to the asymptotic Plateau problem.
result Existence of geodesic lines with Morse index ≤ n-1.
Smooth solutions found for a curvature problem in hyperbolic space.
problem Existence of smooth complete hypersurfaces with prescribed curvature in hyperbolic space.
method Utilized Pogorelov type interior second order estimate.
result Affirmative answers for specific curvature cases in hyperbolic space.
Solves asymptotic An-realization problem for curves.
problem Realization problem for plane curves.
method Asymptotic analysis of smooth An-realization. result Determines cobordism distance between specific knot types.
Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.
problem Minimal submanifolds in negatively curved spaces with small curvature.
method Analysis of spheres at infinity and asymptotic Plateau problem.
result Complete minimal submanifolds bound a class of spheres with uniquely solvable asymptotic Plateau problem.
Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.
problem Boundary behavior of the singular Yamabe problem near singular boundaries.
method Analysis of asymptotic behaviors and derivation of optimal estimates for background metrics.
result Solutions are well approximated by solutions in tangent cones at singular points.
We give a fairly complete solution to the asymptotic Plateau Problem for area minimizing surfaces in H2xR. In particular, we identify the collection of Jordan curves in the asymptotic boundary of H2xR, which bounds an area minimizing surface in H2xR. Furthermore, we study the similar problem for minimal surfaces, and s…
We give a survey on the development of the study of the asymptotic Dirichlet problem for the minimal surface equation on Cartan-Hadamard manifolds. Part of this survey is based on the introductory part of the doctoral dissertation of the author. The paper is organised as follows. First we introduce Cartan-Hadamard mani…
Paper solves Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
problem Solving Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
method Combining covolume, Hadamard variational formula, and geometric interpretation.
result Solved Minkowski problem for non-compact convex sets under asymptotic conditions.
Solves Yamabe problem for Sobolev-class asymptotically hyperbolic manifolds.
problem Solving the Yamabe problem for specific types of asymptotically hyperbolic manifolds.
method Introduces new function spaces and uses Fredholm theorems for elliptic operators.
result Solves the Yamabe problem for asymptotically hyperbolic manifolds with Sobolev-class metrics.
Paper estimates curvature of semi-convex hypersurfaces in hyperbolic space.
problem Estimating curvature of semi-convex hypersurfaces in hyperbolic space.
method Established C2 estimates using a new concavity inequality for hessian equations. result Derived C2 estimates for semi-convex complete hypersurfaces with constant σk curvature. Open problem: fixed-budget best arm identification complexity.
problem Understanding the complexity of identifying the best arm in a fixed budget setting.
method Analyzing existing results and conjectures in the fixed-confidence setting.
result Open questions remain about the fixed-budget setting.
Study on a Bahri-Brezis problem on hyperbolic manifolds.
problem Existence of solutions on asymptotically hyperbolic manifolds.
method Algebraic Topological argument of Bahri-Coron.
result Existence of at least one solution under specific conditions.
Study on spectral asymptotics in elasticity on smooth manifolds.
problem Analyzing spectral asymptotics in linear elasticity on smooth manifolds.
method Established two-term spectral asymptotics for boundary value problems in linear elasticity.
result Corrected erroneous results in previous studies.
Study asymptotic behavior of Weingarten surfaces at infinity.
problem Understanding the behavior of Weingarten surfaces at infinity.
method Derive asymptotic expansion and solve Dirichlet problem.
result Established maximum principle and solved Dirichlet problem.
Study non-fillable curves in a hyperbolic surface with a real line.
problem Non-fillable curves in H2imesR method Asymptotic Plateau problem in H2imesR result First examples of non-fillable finite curves with no thin tail.
New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.
problem Constructing asymptotic convex hypersurfaces in hyperbolic space.
method Approximating hypersurface by geodesic graphs over equidistant hyperplanes.
result Existence of complete, strictly locally convex hypersurfaces with prescribed asymptotic boundary.
Solves area-minimizing surface problem for finite curves in H^2xR.
problem Asymptotic Plateau problem for area-minimizing surfaces.
method Complete solution for finite curves in $\BHH$.
result Fairly complete solution for finite curves in $\BHH$.
This is a survey of old and recent results about the asymptotic Plateau problem. Our aim is to give a fairly complete picture of the field, and present the current situation.
Derives short-term option pricing asymptotics in local-stochastic volatility models.
problem Short-term option pricing in local-stochastic volatility models.
method Large deviations theory and variational methods.
result Explicit series expansions for implied volatility and asymptotic results for European and VIX options.
DETC algorithm achieves asymptotic optimality in multi-armed bandit problems.
problem Multi-armed bandit problem with subgaussian rewards.
method DETC algorithm with two exploration and exploitation phases.
result DETC achieves asymptotically optimal regret bound.
ROOT-SGD solves convex optimization problems with optimal nonasymptotic and near-optimal asymptotic performance.
problem Solving strongly convex and smooth unconstrained optimization problems using stochastic first-order algorithms.
method ROOT-SGD: Recursive One-Over-T SGD, averaging past stochastic gradients.
result Achieves state-of-the-art performance in both nonasymptotic and asymptotic senses.
Develops mixed quantization for graph vector bundles.
problem Solving asymptotic spectral problems on graph vector bundles.
method Mixed quantization technique for graph vector bundles.
result Applications to various spectral problems.
Stochastic approximation proves asymptotic normality for non-smooth problems.
problem Solving non-smooth stochastic approximation problems.
method Stochastic approximation algorithms for solving smooth equations, extended to non-smooth problems.
result Asymptotic normality and optimality in non-smooth stochastic approximation is proven.
We show that if a Jordan curve C in the asymptotic sphere contains a smooth point, there is an embedded H-plane in H^3 asymptotic to C for any H in [0,1).
Paper finds invariant solutions for Plateau problem in hyperbolic space.
problem Finding minimal surfaces with specific symmetries in hyperbolic space.
method Proved existence of foliations by invariant minimal surfaces, used to solve the Plateau problem.
result Existence of invariant minimal surfaces solving the asymptotic Plateau problem.
It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.
Because of the relevance of the results, this paper is merged into the paper titled "On the Number of Solutions to Asymptotic Plateau Problem" (arXiv:math.DG/0505593) as a new section.
Develops asymptotic analysis for RandNLA sampling estimators in least-squares problems.
problem Lack of distributional information for RandNLA estimators in statistical inference.
method Asymptotic analysis of sampling estimators for least-squares problems in two settings.
result Sampling estimators are asymptotically normally distributed under mild conditions.
We prove that a graph G is asymptotically isomorphic to the ray if and only if G is uniformly spherically bounded and is of bounded local degrees. This problem arouse in combinatorics and was posed in [3] (Problem 10.1).
We study the asymptotic behavior of Masur-Veech volumes as the genus goes to infinity. We show the existence of a complete asymptotic expansion of these volumes that depends only on the genus and the number of singularities. The computation of the first term of this asymptotics expansion was a long standing problem. Th…
We show the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recentl…
Study shows how fast a specific matrix completion method works.
problem Completing a rank-one matrix from a subset of revealed entries.
method Alternating minimization approach for matrix completion.
result Polynomial upper bound on convergence rate.
A solution of a problem by V.I.Arnol'd about higher analog of the asymptotic Hopf invariant of divergence-free vector fields is presented. A higher invariant of magnetic fields, which is not expressed from the asymptotic linking numbers of magnetic lines is constructed and examples of an asymptotic invariants is constr…
We create a flat end foliation by critical spheres solving a Laplace-Beltrami problem.
problem Foliation of an asymptotically flat end by critical hypersurfaces.
method Constructing hypersurfaces as critical points of a functional, solving an over-determined boundary value problem.
result Solutions to the Laplace-Beltrami operator over a foliation of critical spheres.
Unified asymptotics for investment in markets with transaction costs and search frictions.
problem Investment in markets with transaction costs and search frictions.
method Power-utility maximization problem with proportional transaction costs and Poisson-triggered trades, analyzed using a novel asymptotic framework.
result Explicit asymptotics for the no-trade region and value function derived.
We consider a non-trapping n-dimensional Lorentzian manifold endowed with an end structure modeled on the radial compactification of Minkowski space. We find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes. The rates of decay seen in the asymptotic expansion a…
Study asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.
problem Asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.
method Sharp expansions derived for the Poisson kernel and Green's functions near singularities.
result Sharp expansions of the Green's functions solve the first part of Kim-Musso-Wei's conjecture.
Solves constant pre-factor problem for tt*-Toda equations using asymptotic data and symplectic structures.
problem Constant pre-factor problem for the tt*-Toda equations.
method Explicit evaluation using asymptotic data and introduction of symplectic structures.
result Preservation of symplectic structures by Riemann-Hilbert correspondence for wider class of solutions.
Study of constant curvature hypersurfaces in hyperbolic space.
problem Finding complete hypersurfaces with constant sum Hessian curvature.
method Solving the asymptotic Plateau problem in hyperbolic space.
result Existence of complete hypersurfaces with specified curvature properties.
In this paper, we study the exterior problem for the maximal surface equation. We obtain the precise asymptotic behavior of the exterior solution at infinity. And we prove that the exterior Dirichlet problem is uniquely solvable given admissible boundary data and prescribed asymptotic behavior at infinity.
We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.
VRPG algorithm optimizes convex constraints with non-asymptotic guarantees.
problem Stochastic convex optimization under convex constraints.
method Natural variance reduced proximal gradient (VRPG) algorithm.
result VRPG achieves local minimax lower bound up to constants and log factor of N. New estimates for Hitchin's equations at high energy.
problem Solutions to Hitchin's self-duality equations at high energy.
method New estimates and asymptotic decoupling phenomenon.
result Generalization to arbitrary Higgs bundles.
We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space E(−1,τ). As one of our main results, we present sufficient conditions for a curve Γ in ∂∞E(−1,τ) to admit a solution to the asymptotic Plateau problem, in the sense th…
Asymptotically optimal algorithm for contextual linear bandits.
problem Contextual linear bandits with suboptimal algorithms.
method Decoupling context distribution and exploration policy, incremental primal-dual approach, confidence intervals.
result Asymptotic optimality and scalability of the algorithm.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
problem Uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
method Analyzes criteria for uniqueness and constructs examples of non-uniqueness.
result Uniqueness of minimal surfaces is equivalent to uniqueness in a smaller class of stable minimal disks.