Study on compact Quasi-Einstein manifolds yields diameter estimates and Hitchin-Thorpe inequality conditions.
problem Estimating diameters and verifying Hitchin-Thorpe inequality for compact Quasi-Einstein manifolds.
method Derive geometric estimates relating potential function oscillation to manifold diameter; derive lower bounds for diameter.
result Diameter conditions ensure compact Quasi-Einstein manifolds satisfy Hitchin-Thorpe inequality in dimension four.
Uniform estimates for Kaehler metrics' diameters and volumes.
problem Estimating diameters and volumes of Kaehler metrics.
method Proving uniform diameter and volume estimates for a family of Kaehler metrics.
result Uniform estimates for diameters and volumes of Kaehler metrics.
Estimates Kaehler metrics' diameter in big cohomology classes.
problem Estimating the diameter of Kaehler metrics in big cohomology classes.
method Proves uniform diameter estimates using integrability conditions and stability properties of complex Monge-Ampere equations.
result Uniform diameter estimates for Kaehler metrics in big cohomology classes.
Estimates graph curvature and diameter using Laplacian eigenvalues.
problem Estimating graph curvature and diameter using Laplacian eigenvalues.
method Combination of gradient estimates and strong nodal domain walks.
result Li-Yau type eigenvalue-diameter estimate for signed graphs.
Estimates submanifold diameters in curved spaces.
problem Estimating the intrinsic diameter of submanifolds in curved spaces.
method Using mean curvature field integrals and boundary lengths.
result Diameter estimates for submanifolds in curved spaces.
Estimates Kähler metric diameters with entropy bound alone.
problem Estimating Kähler metric diameters.
method PDE techniques for L∞ estimates of the Monge-Ampère equation, improving degeneracies. result Diameter bounds for Kähler-Ricci flow and Calabi-Yau manifolds.
New proof for curvature and diameter estimates on Fano manifolds.
problem Curvature and diameter estimates for Kähler-Ricci flow on Fano manifolds.
method New Harnack estimate for special functions in space-time.
result Established new estimates for scalar curvature and diameter.
Upper diameter bound for manifolds with positive scalar curvature.
problem Estimating the maximum size of manifolds with positive scalar curvature.
method Proving an upper diameter bound using scalar curvature integral, Yamabe constant, and manifold dimension.
result The power of scalar curvature integral in diameter estimates is sharp and occurs at round spheres with canonical metric.
Paper estimates diameter for Minkowski problem solutions.
problem Estimating diameter of solutions to Minkowski problem.
method Uniform diameter estimate for Lp dual Minkowski problem. result Uniform diameter estimate for planar Lp dual Minkowski problem. Sharp estimates for mean curvature flow confirm bounded diameter conjecture.
problem Bounding the diameter of mean curvature flow in 3D.
method Quantitative estimates on second fundamental form and singular set.
result Uniform boundedness of intrinsic diameter and sharp estimates on flow properties.
Note removes degeneracy in Kähler geometry estimates.
problem Estimating diameter and inequalities in Kähler geometry with degeneracy.
method Technical improvement of earlier results.
result Established diameter, Green's functions, and Sobolev inequalities without small degeneracy assumption.
Paper bounds surface diameter and solves Plateau-Douglas problem.
problem Bounding the diameter of compact surfaces and solving the Plateau-Douglas problem.
method Geometric argument based on Topping's diameter bound for closed surfaces.
result Explicit nonexistence criterion for the Plateau-Douglas problem.
The paper estimates surface diameter in conformal spaces.
problem Estimating the diameter of surfaces in conformally flat spaces.
method Using mean curvature and boundary length, the paper gives an upper bound for the intrinsic diameter.
result The result provides an a priori estimate for connected solutions of Plateau's problem and a necessary condition for the existence of such solutions.
The study finds a diameter bound for graphs with positive entropic Ricci curvature, with optimal bounds for arithmetic mean.
problem Finding diameter bounds for graphs with positive entropic Ricci curvature.
method Using a localized gradient estimate and an equivalent definition of entropic Ricci curvature, the study derives a Bonnet-Myers type diameter bound.
result The derived diameter bound is optimal for arithmetic mean, but not for logarithmic mean.
We prove a lower bound estimate for the first non-zero eigenvalue of the Witten-Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons. Our results improve some previous estimates which were obtained by the first autho…
In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry-Émery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (J. Differential Geom…
Study on Lin-Lu-Yau curvature and diameter of amply regular graphs.
problem Lower bounds of Lin-Lu-Yau curvature in amply regular graphs.
method Application of Hall's marriage theorem and geometric proof.
result Conference graphs have positive Lin-Lu-Yau curvature.
Study bounds Kähler current diameters on manifolds.
problem Bounding diameters of Kähler currents on manifolds.
method Upper bounds established using Orlicz integrability conditions and Hölder continuity.
result Finite diameter for singular Kähler-Einstein currents with Hölder continuous potentials.
We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein met…
Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.
problem Volume noncollapsing for Kähler metrics induced by complex Monge-Ampère equations.
method Proves local volume noncollapsing estimate with Ricci curvature lower bound.
result Establishes diameter and gradient estimates for Kähler metrics.
Paper extends curvature estimates to new tensor types.
problem Mean curvature and volume comparison estimates for integral generalized quasi-Einstein tensors.
method Extends existing comparison results to new tensor types.
result Global diameter estimates derived from comparison results.
Paper bounds Kähler manifolds' diameter using Orlicz spaces and complex Monge-Ampère equations.
problem Establishing diameter bounds for Kähler manifolds in Orlicz spaces.
method Proving a priori estimates for solutions of complex Monge-Ampère equations in Orlicz spaces using Kołodziej's and Guo-Phong-Tong-Wang's approaches.
result Uniform estimates for Green's function and its gradient for Kähler metrics.
The paper proves diameter bounds and finiteness for amply regular graphs.
problem Proving diameter bounds and finiteness for amply regular graphs.
method Improved curvature estimates and new Bakry-Émery curvature estimates.
result There are only finitely many amply regular graphs with specific parameters.
The paper estimates the diameter of (m,ρ)-quasi Einstein manifolds under specific conditions.
problem Estimating the diameter of (m,ρ)-quasi Einstein manifolds. method Analyzing the properties of (m,ρ)-quasi Einstein manifolds and applying geometric and topological constraints. result An upper bound for the diameter of (m,ρ)-quasi Einstein manifolds is determined. The Kähler-Ricci flow yields bounded diameter and Ricci curvature for minimal models.
problem Estimating the diameter and Ricci curvature of long-time solutions of the Kähler-Ricci flow.
method Analyzing the semi-ample canonical line bundle and using Perelman's estimates.
result Uniform bounds on diameter and Ricci curvature for long-time solutions.
Sharp bounds on diameter and eigenvalues for amply regular graphs.
problem Finding bounds for amply regular graphs' diameter and eigenvalues.
method New ideas relating discrete Ricci curvature to local matching properties, including a novel construction of a regular bipartite graph.
result Sharp diameter and eigenvalue bounds for amply regular graphs.
Paper generalizes Bakry-Émery calculus for curvature and applies to Markov chains.
problem Formulating both Bakry-Émery and entropic curvature simultaneously.
method Generalization of Bakry-Émery calculus, new measure optimality criterion, dimension parameter in entropic curvature.
result Diameter estimates for Markov chains with strictly positive entropic curvature and spectral gap.
New curvature measure for graphs improves diameter and eigenvalue estimates.
problem Estimating properties of graphs using Ricci curvature.
method Introduced integral Ricci curvature Iκ0 for graphs. result Uniform estimates for diameter, number of vertices, and eigenvalue.
The study finds a lower bound for the diameter of gradient ρ-Einstein solitons.
problem Estimating the diameter of gradient ρ-Einstein solitons.
method Using mathematical conditions and properties of solitons to derive a lower bound.
result A lower bound for the diameter of gradient ρ-Einstein solitons is established.
Estimates Laplace eigenvalues and diameter for Lie group metrics.
problem Estimating Laplace eigenvalues and diameter for left-invariant metrics on compact Lie groups.
method Relates left-invariant metrics to positive definite matrices and uses eigenvalue properties.
result Partial answers to Eldredge's conjecture on Laplace eigenvalues and diameter.
Proves estimates for Calabi-Yau metrics as Kahler classes shrink.
problem Estimates for Calabi-Yau metrics under shrinking Kahler classes.
method Proves asymptotic expansion in terms of powers of fiber diameter with uniform C^k-estimates.
result Uniform estimates for all orders of derivatives of Calabi-Yau metrics.
It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.
Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.
problem Proving a fundamental gap lower bound for horoconvex domains in hyperbolic space.
method Reduces the problem to a radial-height problem, compares Dirichlet forms with angular operators, and uses Green estimates.
result Establishes a polynomial \(D^{-3}\) scale fundamental gap lower bound.
The paper extends Bonnet-Myers theorem for manifolds with nonnegative Ricci curvature.
problem Compactness and diameter estimation for manifolds with nonnegative Ricci curvature.
method General curvature conditions for estimating diameter and compactness criteria.
result Established compactness theorems for manifolds with polynomial or exponential Ricci curvature decay.
Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.
problem Estimating the diameter of 3D spaces with non-negative Ricci curvature.
method Proving positive scalar curvature passes to Ricci limit spaces of non-negative curvature.
result Optimal Bonnet-Myers upper bound for 3D spaces.
Given an m-dimensional closed connected Riemannian manifold M smoothly isometrically immersed in an n-dimensional Riemannian manifold N, we estimate the diameter of M in terms of its mean curvature field integral under some geometric restrictions, and therefore generalize a recent work of Topping in the Eucli…
The paper proves geometric comparisons on metric measure spaces with integral Bakry-Émery Ricci tensor bounds.
problem Geometric comparisons on metric measure spaces with specific tensor bounds.
method Integral radial Bakry-Émery Ricci tensor bounds and potential function/gradient bounds.
result Diameter and eigenvalue estimates on smooth metric measure spaces.
Study Fano fibrations and Kähler-Ricci flow singularities, proving diameter bounds and curvature estimates.
problem Analyzing Fano fibrations and Kähler-Ricci flow singularities.
method Developsing a singularity in finite time, using rational initial metrics and collapsing volume forms.
result Diameter bounds and curvature estimates for Fano fibrations and Kähler-Ricci flow singularities.
Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.
problem Understanding 4D generalized Ricci flows with nilpotent symmetry.
method Immortal solutions, type III curvature and diameter estimates, new energy monotonicity.
result Blowdown limits lie in a finite-dimensional family of solutions.
We give an explicit estimate of the area of a closed surface by the diameter and a lower bound of curvature. This is better than Calabi-Cao's estimate for a nonnegatively curved two-sphere.
Uniform Sobolev inequality for Kähler metrics with entropy bound.
problem Establishing Sobolev inequalities for Kähler metrics with entropy bound.
method Uniform Sobolev inequality for Kähler metrics with entropy bound and no lower Ricci curvature bound.
result Derive various geometric estimates for Kähler-Einstein currents.
In this paper, we study the integral curvatures of Finsler manifolds and prove several Myers type theorems.
New method reduces version space for CNNs, improving active learning performance.
problem Sampling bias in active learning hinders optimal hypothesis finding in neural networks.
method Version space reduction through prior mass reduction and diameter reduction, proposing a new Gibbs-vote disagreement method.
result Diameter-based querying method reduces version space more effectively than prior mass reduction and other methods.
Torus covers have controlled volume and diameter under curvature and diameter bounds.
problem Bounding volume and diameter of torus covers with curvature and diameter constraints.
method Using lower Ricci curvature bound and upper diameter bound, constructing finite-sheeted covering spaces.
result Recovering and extending a result of Kloeckner and Sabourau with controlled bounds.
Short proof shows infinite diameter for surface diffeomorphisms.
problem Infinite diameter of surface diffeomorphisms group.
method Short proof using Lp-diameter concept. result Infinite Lp-diameter of Diff0(S,area) group. The study shows how many diameter directions in Besse manifolds relate to Blaschke manifolds.
problem Understanding the relationship between diameter directions and Blaschke manifolds in Besse manifolds.
method Analyzing the properties of Besse manifolds and pinched curvature metrics.
result Besse manifolds with many diameter directions are Blaschke manifolds.
Study on positive scalar curvature and its impact on Ricci limit spaces.
problem The influence of uniformly positive scalar curvature on Ricci limit spaces.
method Investigates uniformly positive scalar curvature on non-collapsed Ricci limit spaces.
result Proves a limit space splits at most n-2 lines or R-factors.
We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp Ln−1-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis…