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.
The diameter function is a topological Morse function.
problem The relationship between systole and diameter functions on Teichmüller space.
method Mapping class group-equivariant topological Morse function approach.
result The diameter function on Teichmüller space is a topological Morse function.
We generalize the maximal diameter sphere theorem due to Toponogov by means of the radial curvature. As a corollary to our main theorem, we prove that for a complete connected Riemannian n n n -manifold M M M having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of …
The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the i…
Sharp upper diameter limit found for Ricci solitons.
problem Bounding the diameter of compact shrinking Ricci solitons.
method Used a sharp logarithmic Sobolev inequality and Vitali-type covering argument.
result Sharp upper diameter bound established in terms of scalar curvature and entropy.
Maximal distortion between geodesic and Euclidean diameters in polygonal domains is studied.
problem Maximal ratio of geodesic to Euclidean diameters in polygonal domains with holes.
method Analyzes convex polygons with holes, using geometric triangulations as a comparison.
result The supremum of the ratio is between Ω ( h 1 / 3 ) Ω(h^{1/3}) Ω ( h 1/3 ) and O ( h 1 / 2 ) O(h^{1/2}) O ( h 1/2 ) for convex polygons. In this paper we study the Weil-Petersson geometry of M g , n ‾ \overline{\mathcal{M}_{g,n}} M g , n , the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of M g , n ‾ \overline{\mathcal{M}_{g,n}} M g , n as a function of g g g and n n n . We show that t…
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.
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…
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.
Infinite diameter proved for contractible loops space.
problem Infinite diameter of contractible loops space in a compact surface.
method New functionals on normed groups, more general than quasi-morphisms.
result Proved infinite diameter of contractible loops space.
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.
We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter. It would follow from a conjecture by Bergeron and Venkatesh that the order of growth in our bound is sharp. We also determine how the numbe…
Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.
problem Bounding the length of shortest closed geodesics on Riemannian manifolds with good covers.
method Generalization of previous results using diameter, volume, and cover elements to bound geodesic length.
result Length of shortest closed geodesic is bounded by a function of volume, diameter, and cover elements.
In the celebrated book entitled Metric Structures for Riemannian and Non-Riemannian Spaces, so-called Green Book, Gromov presented a problem regarding a metric measure space. Gromov posed the question Bound the expansion coefficient from below in terms of the observable diameter. The overall aim of the current study is…
We prove that if a complete connected n n n -dimensional Riemannian manifold M M M has radial sectional curvature at a base point p ∈ M p\in M p ∈ M bounded from below by the radial curvature function of a two-sphere of revolution M ~ \widetilde M M belonging to a certain class, then the diameter of M M M does not exceed that of $\widetild…
The paper estimates the diameter of ( m , ρ ) (m,ρ) ( m , ρ ) -quasi Einstein manifolds under specific conditions.
problem Estimating the diameter of ( m , ρ ) (m,ρ) ( m , ρ ) -quasi Einstein manifolds. method Analyzing the properties of ( m , ρ ) (m,ρ) ( m , ρ ) -quasi Einstein manifolds and applying geometric and topological constraints. result An upper bound for the diameter of ( m , ρ ) (m,ρ) ( m , ρ ) -quasi Einstein manifolds is determined. 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 L p L^p L p -diameter concept. result Infinite L p L^p L p -diameter of D i f f 0 ( S , a r e a ) Diff_0(S,area) D i f f 0 ( S , a r e a ) 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.
An equivalent definition of entropic Ricci curvature on discrete spaces was given in terms of the global gradient estimate. With a particular choice of the density function ρ ρ ρ , we obtain a localized gradient estimate, which in turns allow us to derive a Bonnet-Myers type diameter bound for graphs with positive entropi…
Exact diameter found for some Riemann surfaces.
problem Calculating the diameter of compact Riemann surfaces exactly.
method Proved for a specific class of surfaces (generalized Bolza surfaces).
result Diameters of generalized Bolza surfaces are equal to their fundamental polygon radii.
The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
problem Diameter rigidity of Kähler manifolds with positive holomorphic sectional curvature.
method Establishing diameter rigidity for Kähler manifolds with positive holomorphic sectional curvature.
result Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
Sharp diameter bounds for Calabi-Yau degenerations proved.
problem Bounding the diameter of Calabi-Yau metrics during degeneration.
method Sharp upper and lower bounds derived for Ricci-flat Kahler metrics.
result Conjecture confirmed by obtaining precise diameter bounds.
Sharp bound on smallest diameter of hyperbolic surfaces.
problem Finding the smallest possible diameter of hyperbolic surfaces.
method Proved a specific formula for the minimal diameter.
result Minimal diameter is log ( g ) + 25 log log ( g ) + O ( 1 ) \log(g)+25 \log \log(g) + O(1) log ( g ) + 25 log log ( g ) + O ( 1 ) . Maximal diameter theorem for graphs with positive Ricci curvature.
problem Diameter comparison in directed graphs with positive Ricci curvature.
method Introduced a Lin-Lu-Yau type Ricci curvature for directed graphs and investigated rigidity properties for the equality case.
result Concluded a maximal diameter theorem of Cheng type.
Spheres can be stretched to have larger diameter than antipodal distance.
problem Finding the maximum diameter of spheres relative to antipodal points.
method Deformation of spheres in dimensions ≥3.
result Spheres can be deformed to have diameter larger than antipodal distance.
Max diameter Kahler manifolds with positive bisectional curvature are complex projective spaces.
problem Understanding the rigidity of Kahler manifolds with specific curvature properties.
method Proving isometry to complex projective space using maximal diameter and positive bisectional curvature.
result Kahler manifolds with maximal diameter and positive bisectional curvature are isometric to complex projective spaces.
Makeev proved that among centrally symmetric four-dimensional polytopes, with more than twenty facets and circumscribed about the Euclidean ball of diameter one, there is no universal cover for the family of unit diameter sets. In this paper we examine the converse problem, and prove that each centrally symmetric polyt…
Small sub-Riemannian balls have diameter close to twice their radius.
problem Understanding the diameter of small sub-Riemannian balls.
method Analyzing C 1 , 1 C^{1,1} C 1 , 1 and C 0 C^0 C 0 sub-Riemannian manifolds. result The diameter of small sub-Riemannian balls equals twice the radius in C 1 , 1 C^{1,1} C 1 , 1 manifolds, and is close to twice the radius in C 0 C^0 C 0 manifolds. Extends diameter bounds for submanifolds with boundary and minor curvature restrictions.
problem Bounding the diameter of submanifolds with boundary and minor curvature restrictions.
method Applies bounds dependent on mean curvature and area to minimal, constant mean curvature, and prescribed mean curvature surfaces.
result Diameter bounds for submanifolds with boundary and minor curvature restrictions.
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.
Study on RCD(0,N) spaces with small linear diameter growth.
problem Understanding structure properties of RCD(0,N) spaces.
method Analyzing the (revised) fundamental group of RCD(0,N) spaces.
result Proved that the revised fundamental group is finitely generated for RCD(0,N) spaces with small linear diameter growth.
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.
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.
Study systoles and diameters on hyperbolic surfaces, finding an upper bound for their ratio.
problem Understanding the relationship between systoles and diameters on hyperbolic surfaces.
method Exploring the inequality between systoles and diameters, deducing an upper bound for their ratio.
result The ratio of systoles and diameters has a genus-dependent upper bound.
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.
Weak base-point freeness leads to Kähler-Ricci flow diameter bounds.
problem Bounding the diameter of Kähler-Ricci flow singularities.
method Weak transcendental base-point freeness on Kähler manifolds.
result Diameter lower bound for Kähler-Ricci flow singularities.
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.
Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.
problem Bounding the diameter of Calabi-Yau fibrations near singular fibers.
method Uniform diameter bound proof for Calabi-Yau fibrations with canonical singular fibers.
result Uniform diameter bounds for all fibres in suitable rescaling.
This paper proves a new, more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.
problem Proving a more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.
method Using a combination of Ricci curvature bounds and Riemannian universal cover properties to establish a quantitative rigidity result.
result If a manifold has positive Ricci curvature and a diameter close to the maximal possible, it is diffeomorphic and bi-Hölder close to the sphere.
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.
For an immortal Ricci flow on an m m m -dimensional ( m ≥ 3 ) (m\ge 3) ( m ≥ 3 ) closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a Riemannian orbifold; (2) if the flow is type-III with diameter growth controlled …
In this paper we show that, under some curvature assumptions the integral of distance function on a compact Riemannian manifold is bounded below by the product of diameter, volume and a constant only depending on the dimension.
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 ∞ L^\infty L ∞ estimates of the Monge-Ampère equation, improving degeneracies. result Diameter bounds for Kähler-Ricci flow and Calabi-Yau manifolds.
Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.
problem Determine diameter bounds for Kähler and quaternionic Kähler manifolds under curvature positivity.
method Define orthogonal Bakry-Émery tensor, study diameter theorems, and derive Bonnet-Myers type bounds.
result Sharper diameter bounds than in Riemannian case under specific curvature assumptions.
Uniform diameter bound for reflection group disk patterns.
problem Uniform bounded diameter conjecture for reflection groups.
method Skinning map analysis and discrete extremal width on Coxeter graph.
result Diameter of skinning image is bounded by a constant.