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.
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.
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.
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.
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 …
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.
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. 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.
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.
Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.
problem Finding the minimum value of the first Neumann eigenvalue for convex domains.
method Proved the sharp lower bound using diameter and width.
result Sharp lower bound for the first Neumann eigenvalue established.
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.
Sharp upper bound found for stable minimal surfaces.
problem Bounding the diameter of stable minimal surfaces.
method Analyzing three-dimensional Riemannian manifolds with specific curvature conditions.
result Sharp upper bound for the diameter of stable minimal surfaces.
The paper proves a linear diameter bound for hyperbolic knot complexes.
problem Understanding the diameter of Kakimizu complexes for hyperbolic knots.
method Defined a complex I S ℓ ( K ) IS_\ell(K) I S ℓ ( K ) to study incompressible Seifert surfaces and proved its diameter has a linear upper bound. result The diameter of the Kakimizu complex for hyperbolic knots grows linearly with genus, confirming a conjecture.
We show that the compression body graph has infinite diameter.
New bounds on diameters and generators for specific lattices and graphs.
problem Finding bounds on diameters and generators for arithmetic lattices and Ramanujan graphs.
method Analyzing arithmetic lattices from Eichler orders in quaternion algebras, applying techniques to definite quaternion algebras.
result Bounds on diameters and generators for arithmetic lattices and Ramanujan graphs.
The paper shows mean curvature flow keeps diameter bounded under certain conditions.
problem Proving the bounded diameter of hypersurfaces under mean curvature flow.
method Use of Lojasiewicz inequalities and solution of mean-convex neighbourhood conjecture.
result The intrinsic diameter stays uniformly bounded as the flow approaches the first singular time.
We prove the absence of a universal diameter bound on lengths of curves in a sweep-out of a Riemannian 2-sphere. If such bound existed it would yield a simple proof of existence of short geodesic segments and closed geodesics on a sphere of small 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…
Let G, a subset of O(4), act isometrically on the 3-sphere. In this article we calculate a lower bound for the diameter of the quotient spaces S 3 / G S^3/G S 3 / G . We find it to be 1 / 2 arccos ( tan ( 3 π 10 ) 3 ) {1/2}\arccos(\frac{\tan(\frac{3 π}{10})}{\sqrt3}) 1/2 arccos ( 3 t a n ( 10 3 π ) ) , which is exactly the value of the lower bound for diameters of the spherical space forms. In the p…
The group Diff ( M ) \text{Diff}(\mathcal{M}) Diff ( M ) of diffeomorphisms of a closed manifold M \mathcal{M} M is naturally equipped with various right-invariant Sobolev norms W s , p W^{s,p} W s , p . Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when s p ≤ dim M sp\le \text{dim}\mathcal{M} s p ≤ dim M and s < 1 s<1 s < 1 ). B…
We will present a new proof of the Gromoll-Grove diameter rigidity theorem.
Finite types of 4D manifolds with specific curvature, volume, and diameter.
problem Classifying 4D manifolds with given curvature, volume, and diameter constraints.
method Proving finiteness of diffeomorphism types for 4-manifolds with specified conditions.
result There are only finitely many diffeomorphism types of 4D manifolds with given curvature, volume, and diameter constraints.
We consider the Lie group S U 2 SU_2 S U 2 endowed with a left-invariant axisymmetric Riemannian metric. This means that a metric has eigenvalues I 1 = I 2 , I 3 > 0 I_1 = I_2, I_3 > 0 I 1 = I 2 , I 3 > 0 . We give an explicit formula for the diameter of such metric. Other words, we compute the diameter of Berger's sphere.
Paper estimates diameter for Minkowski problem solutions.
problem Estimating diameter of solutions to Minkowski problem.
method Uniform diameter estimate for L p L_p L p dual Minkowski problem. result Uniform diameter estimate for planar L p L_p L p dual Minkowski problem. The study bounds the effective diameter of graphs with positive Ollivier curvature.
problem Bounding the effective diameter of graphs with positive Ollivier curvature.
method Introducing reflective graphs and proving discrete Bonnet Myers theorem.
result The effective diameter bound is attained only for specific graphs.
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.