Solves a specific Dirichlet problem for Lagrangian mean curvature equations.
problem Solving the Dirichlet problem for Lagrangian mean curvature equations.
method Solves the Dirichlet problem for Lagrangian mean curvature equations on uniformly convex domains.
result Solves the Dirichlet problem for Lagrangian mean curvature equations.
Proves existence of classical Neumann problems for Hessian equations in uniformly convex domains.
problem Existence of solutions to Neumann problems for Hessian equations.
method Proving existence through uniformly convex domains and Alexandrov-Fenchel inequalities.
result Existence of classical Neumann problems for Hessian equations in uniformly convex domains.
Given a hyperbolic domain, the nearest point retraction is a conformally natural homotopy equivalence from the domain to the boundary of the convex core of its complement. Marden and Markovic showed that if the domain is uniformly perfect, then there exists a conformally natural quasiconformal map which admits a bounde…
Proves existence of solution to Neumann problem for Hessian equations.
problem Existence of classical solution to Neumann boundary problem for Hessian equations.
method Established a priori derivative estimates up to second order.
result Affirmative answer to a conjecture by N. Trudinger.
We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the…
Uniform convexity in divisible domains leads to hyperbolic geometry.
problem Understanding the geometry of divisible convex sets in Finsler manifolds.
method Proving β-uniform convexity of a specific Finsler metric. result A strictly convex divisible domain induces a β-uniformly convex Finsler metric. Paper proves approximations and rigidity for minimal surfaces in convex domains.
problem Proving properties of minimal surfaces in convex domains.
method Uniform approximation and rigidity theorems for minimal surfaces.
result Minimal surfaces can be approximated by proper complete immersions.
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.
Flow of convex hypersurfaces in hyperbolic space converges to geodesic spheres.
problem Understanding the evolution of convex hypersurfaces in hyperbolic space.
method Gauss curvature type flow, Alexandrov-Fenchel inequality application.
result Smooth solution converges to geodesic spheres.
This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset G of the Grassmann bundle G(p,TX) of tangent p-planes to a riemannian manifold X. This determines a nonlinear partial differential equation which is convex but never uniformly elliptic (p < dim X). …
Paper finds unique solutions for curved surfaces with specific gradient.
problem Existence of curved surfaces with specific gradient.
method Second boundary value problem of constant mean curvature equations.
result Unique convex solutions for constant mean curvature equations.
Let Ωand \tildeΩ be uniformly convex domains in \mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: Ω\to \tildeΩ such that the graph Σ= \{(x,f(x)): x \in Ω\} is a minimal Lagrangian submanifold of \mathbb{R}^n \times \mathbb{R}^n.
We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smoot…
Groups with specific subgroup actions have proper actions on uniformly convex spaces.
problem Proper actions of hyperbolic relative groups on Banach spaces.
method Affine isometric actions on uniformly convex Banach spaces.
result Groups with proper actions on uniformly convex spaces.
Compactness proven for manifolds with nonnegative Ricci curvature and uniformly convex boundary.
problem Compactness of manifolds with specific curvature and boundary conditions.
method Monotone quantities constructed from positive proper harmonic functions with Neumann condition.
result Proves compactness of manifolds with nonnegative Ricci curvature and uniformly convex boundary.
For a domain Ω⊂Rn, we introduce the concept of a uniformly Cm defining function. We characterize uniformly Cm defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly Cm defining functions. Some of ou…
2-convex translating solitons are locally strictly convex.
problem Characterizing the convexity of translating solitons in mean curvature flow.
method Analyzing uniformly 2-convex translating solitons in Rn+1. result Locally strictly convex translating solitons are axisymmetric.
New method turns optimization algorithms into uniformly stable learning algorithms for non-Euclidean norms.
problem Non-Euclidean norms in binary classification problems.
method Black-box reduction method using uniformly convex regularizers.
result Achieves optimal statistical risk bounds on excess risk for non-Euclidean norms.
This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.
Unified methods solve convex and nonconvex optimization problems.
problem Solving nonlinear programming problems, convex and nonconvex.
method Incorporating local search steps into uniformly optimal convex programming methods.
result Achieve best known complexity for nonconvex problems and optimal for convex ones.
New proof shows symmetry for certain curved surfaces in higher dimensions.
problem Understanding symmetries in curved surfaces evolving over time.
method Generalized previous result to higher dimensions, proving symmetry condition.
result Uniformly 3-convex translating solitons must have SO(n−1) symmetry. Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
The paper proves inequalities and consequences in Hilbert metrics and Hitchin representations.
problem Volume entropy rigidity and length spectrum comparison in specific geometric settings.
method Sharp distance inequalities and volume growth analysis.
result Volume entropy rigidity for Hilbert geometries and length spectrum comparison for Hitchin representations.
Classifies 3-manifolds with uniformly positive scalar curvature.
problem Classifying 3-manifolds with uniformly positive scalar curvature.
method Analyzes properties of 3-manifolds with mean convex boundaries and uniformly positive scalar curvature.
result 3-manifolds with uniformly positive scalar curvature are homeomorphic to sums of spherical 3-manifolds and S1imesS2. Study shows linear growth of index for free boundary minimal hypersurfaces.
problem Understanding the index growth of free boundary minimal hypersurfaces.
method Analyzes the Morse index growth in relation to homology groups and boundary components.
result Linear growth of index with dimension of first relative homology group and number of boundary components.
The paper introduces a new flow to converge to a Wulff shape from a smooth convex hypersurface.
problem Proving Alexandrov-Fenchel inequalities for anisotropic mixed volumes.
method Introducing a fully nonlinear locally constrained anisotropic curvature flow.
result The flow converges smoothly and exponentially to a scaled Wulff shape.
Study shows Stochastic Mirror Descent optimizes convex problems with infinite noise variance.
problem Optimizing convex problems with infinite noise variance.
method Stochastic Mirror Descent algorithm with uniformly convex mirror maps.
result Demonstrates convergence rate quantified in terms of iterations, dimensionality, and geometric parameters.
Constructs uniformly positive scalar curvature metrics on open manifolds
problem Finding uniformly positive scalar curvature metrics on open manifolds
method Using Morse functions and exhaustion
result Proving the existence of uniformly positive scalar curvature metrics
We study strip deformations of convex cocompact hyperbolic surfaces, defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface. We prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation, in…
New algorithm uniformly samples high-dimensional convex bodies efficiently.
problem Uniform sampling of high-dimensional convex bodies.
method Stochastic diffusion perspective to show contraction to the target distribution.
result Achieves state-of-the-art runtime complexity with strong guarantees on output.
The paper studies curvature flows in hyperbolic space and proves convergence to spheres under certain conditions.
problem Curvature flows in hyperbolic space and their convergence properties.
method Analyzes a class of flows with specific speed functions and proves convergence under various conditions.
result The mean convex and uniformly convex solutions to the flow converge to spheres for specified conditions.
Let U⊆Rn be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function f:U→R can be approximated by real analytic convex functions, uniformly on all of U. In doing so we provide a technique which transfers results on uniform approximation on bounded …
Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklo…
Affine deformations of convex cones yield special spacetime structures.
problem Deforming divisible convex cones in affine spaces.
method Analyzing the maximal convex domains and quotient structures.
result Quotients of affine actions are MGHCC affine spacetimes.
In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the ε-covering number of $\C([a, b]^d, B)$, in the Lp-metric, 1≤p<∞, in terms of the relevant constants, where d≥1, $a < b \in \mathb…
Unified approach for first-order methods with Markovian noise in stochastic optimization and variational inequalities.
problem Stochastic optimization problems with Markovian noise.
method Unified theoretical analysis of first-order gradient methods using randomized batching and multilevel Monte Carlo.
result Optimal (linear) dependence on the mixing time of the noise sequence, eliminating previous limiting assumptions.
The paper studies a flow of convex hypersurfaces expanding by their support and curvature functions.
problem Analyzing the behavior of expanding hypersurfaces in Euclidean space.
method Introduced a curvature flow with specific speed function and proved the existence and convergence of the flow under certain conditions.
result The flow converges to a round sphere centered at the origin for all time under specific conditions.
GenFlow optimizes faster, avoiding saddle points in fixed time.
problem Designing efficient optimization algorithms for convex and non-convex functions.
method Introduces GenFlow and momentum variants with fixed-time convergence guarantees.
result GenFlow and momentum variants converge to optimal solutions in fixed time for PL functions and evade saddle points uniformly.
We show that for acylindrically hyperbolic groups Γ (with no nontrivial finite normal subgroups) and arbitrary unitary representation ρ of Γ in a (nonzero) uniformly convex Banach space the vector space Hb2(Γ;ρ) is infinite dimensional. The result was known for the regular representations on ℓp(Γ) with …
New algorithms optimize convex functions with high-order derivatives.
problem Optimizing convex functions with high-order derivatives under various norms.
method Developed a non-Euclidean inexact accelerated proximal point method using an inexact uniformly convex regularizer.
result Showed nearly optimal algorithms for high dimensions in the black-box oracle model for ℓp-settings and all q≥1. Paper establishes tight lower bounds for minimizing certain smooth and convex functions.
problem Minimizing high-order Hölder smooth and uniformly convex functions.
method Analyzes two asymmetric cases of q>p+ν and q<p+ν using worst-case oracle complexities. result Establishes worst-case oracle complexities for reaching an ε-approximate solution.
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.
New method approximates short immersions as C^{1,θ} isometric immersions for n ≥ 3.
problem Constructing C^{1,θ} isometric immersions of Riemannian metrics.
method Convex integration scheme with iterative integration by parts procedure.
result Uniform approximation of any short immersion by C^{1,θ} isometric immersions for θ < 1/(1+2(n-1)).
New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.
problem Analyzing convex co-compact hyperbolic 3-manifolds with compressible boundaries.
method Defining and analyzing a new version of the renormalized volume.
result The adapted renormalized volume is bounded and has properties analogous to the classical renormalized volume.
Classifies ancient convex curves in convex domains.
problem Ancient convex curve shortening flows on convex domains.
method Classification of convex ancient solutions.
result Ancient convex curves in convex domains classified.
In this paper we prove the probabilistic continuous complexity conjecture. In continuous complexity theory, this states that the complexity of solving a continuous problem with probability approaching 1 converges (in this limit) to the complexity of solving the same problem in its worst case. We prove the conjecture ho…
We prove that any translating soliton for the mean curvature flow which is noncollapsed and uniformly 2-convex must be the rotationally symmetric bowl soliton. In particular, this proves a conjecture of White and Wang, in the 2-convex case in arbitrary dimension.
A new framework optimizes model transfer across domains with labeled data.
problem Distributional heterogeneity across domains in multi-source learning.
method Conditional Group Distributionally Robust Optimization (CG-DRO) framework with Mirror Prox algorithm and double machine learning.
result Established fast statistical convergence rates and uniformly valid inference for CG-DRO.