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.
Small sub-Riemannian balls have diameter close to twice their radius.
problem Understanding the diameter of small sub-Riemannian balls.
method Analyzing C1,1 and C0 sub-Riemannian manifolds. result The diameter of small sub-Riemannian balls equals twice the radius in C1,1 manifolds, and is close to twice the radius in C0 manifolds. Connectedness of small clusters in Riemannian and Finsler manifolds proven.
problem Understanding connectedness of small clusters in Riemannian and Finsler manifolds.
method Proved connectedness and small diameter properties for clusters of small volume in both manifolds.
result Clusters in Riemannian manifolds are connected and have small diameter; in Finsler manifolds, they are at most m connected components of small diameter.
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.
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.
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.
Metric surfaces can be divided into small triangles.
problem Decomposing metric surfaces into triangles.
method Proving any metric space homeomorphic to a surface can be divided into non-overlapping convex triangles of small diameter.
result Metric surfaces can be decomposed into triangles of arbitrarily small diameter.
Simple curves enclose two small disks if they're wide and bend moderately.
problem Bounding the diameter of a curve to enclose two disjoint unit disks.
method Analyzing curvature and diameter constraints of a simple closed curve.
result A curve with curvature ≤1 and diameter ≥4 encloses two disjoint open unit disks.
Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M,g)=1 and every 1-cycle dividing M into two regions of equal area has length >C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Ste…
Proves existence of maps with controlled small curvatures.
problem Existence of locally distance-increasing maps with controlled curvatures.
method Proves existence using controlled small curvatures.
result Existence of locally distance-increasing maps with controlled small curvatures.
Negative curvature restricts the gap between the first and second eigenvalues of convex domains.
problem The fundamental gap of convex domains is limited by negative curvature.
method Adapted from Bourni et. al. (2022) for Riemannian manifolds with negative sectional curvature.
result The product of the fundamental gap and the square of the diameter can be arbitrarily small in domains with negative curvature.
Fundamental gap vanishes for convex domains in hyperbolic space.
problem Behavior of fundamental gap in convex domains in hyperbolic space.
method Proof for Laplace operator with Dirichlet boundary conditions.
result Fundamental gap can be arbitrarily small for domains of any diameter.
In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fund…
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.
Tight embeddings of 2-tori in 3D space contain short loops.
problem Finding the shortest non-contractible loops in twisted 2-tori.
method Proving systolic inequalities for T2 embeddings in R3. result Highly twisted 2-tori contain non-contractible loops of small diameter.
Surfaces in 3-manifolds concentrate at curvature critical points.
problem Understanding concentration of surfaces in 3-manifolds.
method Proving surfaces concentrate at critical points of scalar curvature.
result Simply connected H-surfaces concentrate at curvature critical points.
The paper proves inequalities for submanifolds in Riemannian manifolds.
problem Proving geometric inequalities for submanifolds in Riemannian manifolds.
method Using Rauch's comparison theorem and first variation formula.
result General Li-Yau inequality applicable in bounded sectional curvature manifolds.
In this paper, we show that, under arbitrary bounded Willmore energy assumption, embedded Willmore spheres (or more generally, embedded Willmore spheres under area constraint) with small diameter in a given 3-dimensional Riemannian manifold (M,h) necessarily concentrate at a critical point of the scalar curvature …
This paper studies graph curvature and its geometric implications.
problem Analyzing non-constant Ricci curvature bounds on graphs.
method Proves eigenvalue estimates, finiteness of fundamental group, diameter bounds, Harnack inequality, and Buser inequality under specific curvature conditions.
result Establishes spectral positive Bakry-Émery Ricci curvature on graphs, providing new geometric insights.
We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions.
We analyze the limit of the p-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence.
We consider the setting of linear regression in high dimension. We focus on the problem of constructing adaptive and honest confidence sets for the sparse parameter θ, i.e. we want to construct a confidence set for theta that contains theta with high probability, and that is as small as possible. The l_2 diameter of a …
Let f:S1→G be a surjective map from the standard unit circle to a graph G such that the pre-image of each point has diameter less than ε. If ε is small enough, does f split as a free factor in π1(G)?
We derive a bound on the L∞-norm of the covariant derivative of Laplace eigensections on general Riemannian vector bundles depending on the diameter, the dimension, the Ricci curvature of the underlying manifold, and the curvature of the Riemannian vector bundle. Our result implies that eigensections with sma…
Inspired by a recent work of Grove-Petersen in [GP18], where the authors studied Alexandrov spaces with largest possible boundary. We study Alexandrov spaces with lower curvature bound 1 and with small boundary. When the radius of X is π/2, and the boundary has diameter π/2, we classify the total space X.
Study axisymmetric ideal fluids on 3-manifolds, proving Fredholm properties.
problem Riemannian geometry of axisymmetric ideal fluids.
method Proving Fredholm properties of L2 exponential map for axisymmetric flows. result Axisymmetric diffeomorphisms form a totally geodesic submanifold.
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.
New connections found between curvature and Euler characteristic using Schrödinger operators.
problem Establishing relationships between curvature and topological invariants of Riemannian manifolds.
method Using twisted Dirac operators and scaling of potentials to analyze the kernel of these operators.
result Found conditions under which the Euler characteristic of a manifold can be zero or non-zero.
Sharp Sobolev inequality derived for Riemannian manifolds with bounded Ricci curvature.
problem Deriving a sharp Sobolev inequality for Riemannian manifolds with bounded Ricci curvature.
method Reduction to functions with small volume support, first order uniform asymptotic expansion of isoperimetric profile, local uniform Sobolev inequality.
result Sharp Sobolev inequality for W1,p(M) into Ln−pnp(M) is derived. 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)+25loglog(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.
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.
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…
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.
We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower…
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-manifold M having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of …
The paper studies the diameter of diffeomorphism groups with Sobolev metrics.
problem Determine the diameter of diffeomorphism groups with right-invariant Sobolev metrics.
method Analyzes various right-invariant Sobolev norms and their effects on the geodesic distance.
result The diameter of the diffeomorphism group is infinite for strong enough norms and finite for weak enough norms.