Sharp criterion for Chern-Gauss-Bonnet integral using Q curvature.
problem Quantifying the Chern-Gauss-Bonnet integral using Q curvature.
method New approach involving singular integral estimation.
result Derivation of asymptotic formula for Q curvature equation.
Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.
problem Relating Yamabe invariants on asymptotically Poincare-Einstein manifolds.
method Established a sharp inequality using lower Ricci curvature bounds.
result Sharp inequality relating type II Yamabe invariant of the interior to the Yamabe invariant of the conformal infinity.
Sharp bounds derived for eigenvalues on specific geometric spaces.
problem Eigenvalue problems on asymptotically hyperbolic manifolds and submanifolds.
method Sharp bounds derived for three types of eigenvalue problems: p-Dirichlet, polyharmonic, and weakly Poincaré-Einstein. result Sharp bounds and their implications for asymptotic sectional curvatures and mean curvature.
We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.
Sharp inequality found for hypersurfaces in curved spaces.
problem Establishing geometric inequalities for hypersurfaces in curved spaces.
method Standard comparison methods in Riemannian Geometry.
result Sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature.
Sharp inequalities for J functional on Kahler metrics.
problem Asymptotic behavior of J functional on Kahler metrics. method Sharp inequalities derived for d1 metric. result Sharp inequalities between J functional asymptotics. Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be esti…
Sharp curvature estimates for mean curvature flow in spheres.
problem Understanding the behavior of surfaces evolving under mean curvature flow in spheres.
method Proving asymptotically sharp curvature pinching estimates and using them to derive derivative and convexity estimates.
result Partial classification of singularity models and new rigidity results for ancient solutions.
Sharp inequalities for manifolds with nonnegative curvature.
problem Establishing inequalities for manifolds with nonnegative curvature.
method Using the ABP-method, generalizing previous work by Brendle.
result Sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds.
Derives new monotone quantities for p-harmonic functions on asymptotically flat 3-manifolds.
problem Estimating the mass of 3-manifolds with non-negative scalar curvature and minimal boundary.
method Derives monotone quantities for p-harmonic functions and applies them to derive a sharp mass-capacity estimate.
result Derives a sharp mass-capacity estimate relating the ADM mass of a 3-manifold to the p-capacity of its boundary.
Sharp Sobolev and Michael-Simon inequalities on curved manifolds.
problem Proving inequalities on manifolds with nonnegative curvature.
method Analyzing asymptotic volume ratio and curvature properties.
result Sharp Sobolev and Michael-Simon inequalities established.
New hyperbolic manifolds show exponential homology torsion growth.
problem Understanding growth of torsion in homology groups of hyperbolic manifolds.
method Constructed a family of hyperbolic manifolds with specific growth properties.
result Demonstrated that recent bound on homological torsion is asymptotically sharp.
Proves a sharp inequality for toroidal surfaces in Horowitz-Myers geon.
problem Extending Minkowski inequality to non-static spacetimes.
method Proves a sharp inequality for toroidal hypersurfaces in Horowitz-Myers geon.
result Sharp inequality for toroidal surfaces in asymptotically hyperbolic manifold.
We study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector. When the set of unit vectors …
Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.
problem Proving logarithmic Sobolev inequalities on manifolds with nonnegative curvature.
method Employing the ABP method developed by Brendle.
result Sharp L2 and Lp logarithmic Sobolev inequalities established. 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.
Paper proves flat 3-manifolds with positive mass have unique isoperimetric surfaces.
problem Finding unique isoperimetric surfaces in flat 3-manifolds.
method Used 'fill-in' argument and sharp isoperimetric inequality.
result Each leaf of the canonical foliation is the unique isoperimetric surface.
Improved mass-capacity bounds for specific 3D manifolds.
problem Sharp mass-capacity inequality and upper bounds for 3D asymptotically flat manifolds.
method Monotonicity formulas associated with a harmonic potential.
result Improved bounds on ADM mass and capacity in terms of boundary area.
New flexible confidence sequences for robust statistical inference.
problem Creating robust statistical inference methods that work under mild assumptions.
method Proposed a new class of asymptotic time-uniform confidence sequences.
result Sharp asymptotic time-uniform confidence sequences achieved under mild assumptions.
The study examines asymptotic properties of G2-monopoles on nonparabolic G2-manifolds.
problem Investigating the asymptotic behavior of G2-monopoles on nonparabolic G2-manifolds.
method Proof of finite mass for monopoles with bounded curvature on nonparabolic G2-manifolds; sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
result Sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
Sharp asymptotics reveal how network width controls learnability in quadratic neural networks.
problem Understanding learnability in overparameterized quadratic neural networks.
method Mapping ERM to convex matrix sensing with nuclear norm penalization.
result Characterization of global minima and precise generalization thresholds.
Sharp bound on scalar curvature integral in 3-manifolds.
problem Bounding the integral of scalar curvature on 3-manifolds.
method Geodesic ball analysis with nonnegative Ricci curvature.
result Integral of scalar curvature is bounded by 8πR for large radii. We first investigate the asymptotics of conical expanding gradient Ricci solitons by proving sharp decay rates to the asymptotic cone both in the generic and the asymptotically Ricci flat case. We then establish a compactness theorem concerning nonnegatively curved expanding gradient Ricci solitons.
Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.
problem Proving sharp Sobolev inequalities on noncompact Riemannian manifolds with non-negative Ricci curvature.
method Using Optimal Mass Transportation with quadratic distance cost.
result Sharp Lp-Sobolev and Lp-logarithmic Sobolev inequalities established for p>1 and p=1. We prove a comparison theorem on the first Neumann eigenvalue on Bakry-Emery manifolds. Examples are constructed to illustrate the sharpness of the result. A linear explicit lower bound is also proved. We also discuss the asymptotic sharpness of such a result.
The paper explores sharp isoperimetric properties on non-compact spaces with Ricci bounds.
problem Sharp isoperimetric properties on non-compact spaces with Ricci bounds.
method Sharp isoperimetric comparison theorems and asymptotic isoperimetric properties.
result Almost regularity theorems and enhanced functional inequalities.
Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.
problem Estimating asymptotic metrics in 2-step nilpotent Lie groups.
method Developed a novel technique to perturb rectifiable curves.
result Every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone.
Sharp 3D Alexandrov inequality applied to volume-preserving flows.
problem Volume-preserving geometric flows in 3D space.
method Sharp quantitative Alexandrov inequality for C2-regular sets. result Established a 3D sharp quantitative version of the Alexandrov inequality.
Sharp inequalities for matrix means with unknown variance.
problem Estimating matrix means with unknown variance.
method Empirical Bernstein inequalities for symmetric random matrices.
result Adapts to unknown variance with tight deviation bounds.
The paper develops new methods to study sharp isoperimetric properties on complex spaces.
problem Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds.
method Original argument to estimate first and second variation of the area for isoperimetric sets, avoiding regularity theory.
result Generalizes results for smooth and non-compact manifolds, Alexandrov spaces, and convex bodies.
Minimal surfaces in a ball have limited area.
problem Bounding the area of genus zero minimal surfaces in a unit ball.
method Proving an area inequality and showing convergence of saturating sequences.
result The area of each nonflat surface is less than its radial projection, with sharp asymptotic bounds.
The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin-Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic n…
Sharp eigenvalue estimates for submanifolds of asymptotically hyperbolic spaces.
problem Estimating eigenvalues of the p-Laplacian on submanifolds of asymptotically hyperbolic manifolds.
method Sharp upper and lower bounds derived using conformal techniques and properties of submanifolds.
result Lower bounds on the first eigenvalue for minimal and bounded mean curvature submanifolds.
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.
Sharp decay found for solutions of a specific equation in Lie groups.
problem Asymptotic decay of solutions to a Yamabe type equation.
method Analysis of a specific pseudodifferential operator in a homogeneous Lie group.
result Established sharp asymptotic decay of positive solutions.
Prove rigidity and classification results for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
problem Quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
method Prove rigidity and classification results for the quasilinear Liouville equation associated with the n-Laplacian on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. result Under a sharp logarithmic lower bound, the ambient manifold must be isometric to the Euclidean space and the solution must be one of the standard bubbles.
We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.
Sharp boundaries for detecting dense subhypergraphs established.
problem Detecting dense subhypergraphs in random hypergraphs.
method Established sharp detection boundaries for known and unknown edge probabilities.
result Sharp detectable regions differ significantly from graph counterparts.
Sharp bounds derived for test error of finite-rank kernel ridge regression.
problem Loose bounds on test error for finite-rank kernels in machine learning.
method Sharp non-asymptotic upper and lower bounds for KRR test error.
result Tighter bounds on finite-rank KRR test error, valid for any regularization parameters.
Sharp bounds for high-probability estimation of discrete distributions.
problem Estimating discrete distributions with high probability under χ2-divergence. method Sharp upper and lower bounds for the classical Laplace estimator, and characterization of minimax high-probability risk for any estimator.
result Sharp bounds for high-probability estimation of discrete distributions can be achieved through a simple smoothing strategy.
Sharp inequalities and extremizers for J functional on Kähler manifolds.
problem Understanding the large scale asymptotic of the J functional on Kähler metrics.
method Proving sharpness of inequalities and studying extremizing potentials/rays on toric Kähler manifolds, and existence of radial extremizers on general Kähler manifolds.
result Sharpness of inequalities and existence of extremizing potentials/rays on toric Kähler manifolds, and equivalence with plurisupported currents on general Kähler manifolds.
Sharp stability of isometries on Heisenberg group proven.
problem Quantitative stability of isometries on the Heisenberg group.
method Proving quasi-isometries close to isometries with specific closeness orders.
result Quasi-isometries of Heisenberg group are close to isometries with specific closeness orders.
Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.
problem Establishing bounds for eigenfunctions on product spaces.
method Combining asymptotics of Jacobi polynomials and positivity of Fourier coefficients of spherical functions.
result Sharp Lp bounds for eigenfunctions on products of rank-one symmetric spaces. New proof shows D-SGD and SAM are equivalent, revealing advantages of decentralization.
problem The generalization benefits of decentralized learning.
method Proved D-SGD implicitly minimizes SAM's loss function.
result Decentralized SGD and Average-direction SAM are asymptotically equivalent.
The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an …
We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.
We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial su…
Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.
problem Estimating spectral gaps for higher-order operators on Cartan-Hadamard manifolds.
method Symmetrization-free proofs based on general functional inequalities.
result Solves a sharp asymptotic problem from Cheng and Yang and answers a question from Kristály.