We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the b…
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168,742 papers · 148 categories
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48 results for “positive curvature gap”
A new topological gap theorem improves the systole of 3-manifolds with positive scalar curvature.
problem Improving the systole of 3-manifolds with positive scalar curvature.
method Weak inverse mean curvature flow.
result The systole of 3-manifolds is no greater than an improved constant c ≈ 5.44π.
The study improves fundamental gap estimates for surfaces with non-constant positive curvature.
problem Estimating the fundamental gap for surfaces with non-constant positive curvature.
method Using a two-point maximum principle, the study establishes log-concavity and fundamental gap estimates.
result Corresponding log-concavity and fundamental gap estimates for surfaces with non-constant positive curvature are derived.
A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this …
New theorem shows curvature concentration depends linearly on volume ratio.
problem Gap theorem for nonnegative Ricci curvature manifolds with small curvature concentration.
method Exhibited Ricci flow solution with faster than 1/t curvature decay.
result Curvature concentration depends linearly on asymptotic volume ratio.
In this paper, we prove a gap result for a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative Ricci curvature and a scalar curvature average condition. We show that if it has positive Green function, then it is flat. This result is proved by setting up new global Yamabe flow. Ot…
Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.
problem Proving log-concavity of eigenfunctions on curved surfaces.
method Analyzing the Laplacian eigenfunctions on positively curved surfaces.
result Strong log-concavity of the first eigenfunction on positively curved surfaces.
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.
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.
The paper proves parabolic gap theorems for Yang-Mills energy.
problem Yang-Mills energy and instantons on various manifolds.
method Parabolic Yang-Mills flow and Morrey norms.
result Spaces of connections with Yang-Mills energy less than a certain threshold deformation-retract onto spaces of instantons.
In the first part we use Gromov's K--area to define the K--area homology which stabilizes into singular homology on the category of pairs of compact smooth manifolds. The second part treats the questions of certain curvature gaps. For instance, the --curvature gap of complex vector bundles on a compact manif…
An optimal gap theoremmath.DG
By solving the Cauchy problem for the Hodge-Laplace heat equation for -closed, positive -forms, we prove an optimal gap theorem for Kähler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius centered at any f…
Master thesis proves Bergman kernel asymptotics for positive line bundles.
problem Proving asymptotic expansion of Bergman kernel for positive line bundles.
method Introduced a semi-classical symbol space and symbolic calculus.
result Established pointwise asymptotic expansion on positive parts of certain semi-positive line bundles.
The study confirms a conjecture about Kähler manifolds with quasi-negative curvature.
problem Confirming a long-standing conjecture about Kähler manifolds with quasi-negative curvature.
method Introducing --quasi-negativity and applying gap-type theorems.
result Obtained gap-type theorems for in terms of real bisectional curvature and weighted orthogonal Ricci curvature.
Motivated by a previous work of Zheng and the second named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular we prove a gap theorem following the work of Petersen and Tao on Riemannian manifolds with almost quarter-pinched sectional curvature.
Ricci flow controls curvature on manifolds with bounds.
problem Controlling curvature on manifolds with given bounds.
method Ricci flow with curvature bounds and entropy controls.
result Global curvature control at positive times for manifolds.
In this paper, by applying a linear trace Li-Yau-Hamilton inequality for a positive (1,1)-form solution of the CR Hodge-Laplace heat equation and monotonicity of the heat equation deformation, we obtain an optimal gap theorem for a complete strictly pseudocovex CR manifold with nonnegative pseudohermitian bisectional c…
A Uniformization Theorem Of Complete Noncompact Kähler Surfaces With Positive Bisectional Curvaturemath.DG
In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold of positive and bounded holomorphic bisectional curvature, suppose its…
The study proves a gap theorem for CAT(0) spaces with a constant below 1/(6√π).
problem Proving isoperimetric inequalities in non-positive curvature spaces.
method Introduced minimal tetrahedra to prove a linear inequality.
result Established a gap theorem for CAT(0) spaces with a constant below 1/(6√π).
As was recently observed by M. Xu and J. Wolf, there is a gap in Berard Bergery's classification of odd dimensional positively curved homogeneous spaces. Since this classification has been used in other papers as well, we give a modern, complete and self contained proof (in odd as well as even dimensions), confirming t…
Paper proves Simon's third gap conjecture for minimal surfaces in spheres.
problem Investigating the third gap problem in Simon's conjecture for minimal surfaces in unit spheres.
method Developed refined third-order Simons-type integral identities and established new lower bounds for curvature terms.
result Obtained positive gap results for the squared norm of the second fundamental form throughout the interval \(\left[\frac{5}{3},\frac{9}{5}
ight]\).
We prove an energy gap result for Yang-Mills connections on principal -bundles over compact Kähler surfaces with positive scalar curvature. We prove related results for compact simply-connected Calabi-Yau -folds.
Proves pinched Ricci curvature conjecture in all dimensions.
problem Pinched Ricci curvature conjecture in complete non-compact manifolds.
method Develops a lifting technique to handle collapsed manifolds and proves a Ricci flow curvature estimate.
result Direct analogue of Hamilton's result in all dimensions.
A gap in the proof of the main result in reference [1] in our original submission propagated into the constructions presented in the first version of our manuscript. In this version we give an alternative proof for the existence of Riemannian metrics with positive Ricci curvature on an infinite subfamily of closed, sim…
Study gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.
problem Understanding gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.
method Analyzes high tensor powers of Hermitian line bundles with non degenerate curvature, proving Riemann-Roch numbers for eigenvalue clusters and describing spectral projectors.
result Clusters and gaps in eigenvalues are described by Riemann-Roch numbers and have pointwise kernel descriptions.
The paper improves -estimates for Dirac-Dolbeault operators on complex manifolds.
problem Improving -estimates for Dirac-Dolbeault operators on complex manifolds.
method Generalized classical method to handle mixed curvature cases and provided bounds on error terms.
result Full asymptotic expansion for Bergman kernel obtained.
Study on Ricci flow on 4-spheres, proving standard sphere convergence.
problem Characterizing and understanding Ricci flow on 4-spheres.
method Investigation of integral conformal invariants, analysis of flow properties.
result Established monotonic decay of certain curvature norms, leading to standard sphere convergence.
The paper explores gaps in curvature-related metrics and rigidity.
problem Understanding gaps in curvature-related metrics and rigidity.
method Analyzes three types of gaps: spectral, metric-rigidity, and topological-rigidity.
result Proposes open problems in the field.
The paper provides uniform length estimates for trajectories on flat cone surfaces.
problem Estimating the length of trajectories on flat cone surfaces.
method Using self-intersection numbers and constants depending only on the flat metric, the paper focuses on convex flat cone spheres with a positive curvature gap and a fixed number of singularities.
result Uniform two-sided estimates for trajectory lengths on convex flat cone spheres are obtained.
The paper proves gap theorems for Yang-Mills on manifolds with positive Yamabe.
problem Yang-Mills theory on manifolds with positive Yamabe constant.
method Extending Gursky-Kelleher-Streets results to complete manifolds.
result Equality in gap theorem described in terms of basic instanton.
Study on self-similar solutions of supercritical Fujita equation, proving entropy and energy gap.
problem Characterization and stability of solutions to supercritical Fujita equation.
method Introduction of -functional, -stability, and entropy; use of mean curvature flows.
result Constant solution has lowest entropy among bounded positive self-similar solutions.
We consider a rigidity problem for the spectral gap of the Laplacian on an -space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive . For a weighted Riemannian manifold, Cheng--Zhou showed that the sharp spectral gap is achieved only when a -dimensional G…
Kahler-Einstein metrics linked to eigenvalue gaps on Fano manifolds.
problem Existence of Kahler-Einstein metrics on Fano manifolds.
method Characterization via eigenvalue gaps of Cauchy-Riemann and Hamiltonian vector fields.
result Existence of Kahler-Einstein metrics linked to eigenvalue gaps.
Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
Sharp Hardy and spectral gap inequalities found on special irreversible Finsler manifolds.
problem Understanding Hardy and spectral gap inequalities on irreversible Finsler manifolds.
method Finslerian extension of the method of Riccati pairs.
result Sharpness of Hardy and spectral gap inequalities on specific Finsler manifolds.
The study finds larger gaps in mean curvature for biharmonic submanifolds in spheres.
problem Understanding gaps in mean curvature for biharmonic submanifolds.
method Analyzing proper biharmonic submanifolds with parallel mean curvature vector field in Euclidean spheres.
result Determining larger gaps in mean curvature for specific biharmonic submanifolds.
Study on 4D solitons with curvature constraints.
problem Characterizing gradient shrinking Ricci solitons with positive modified sectional curvature.
method Sharp pinching conditions, weighted integral gap results, Hitchin-Thorpe inequality.
result Locally Kähler property under specific curvature conditions.
Study gap phenomenon in flat manifolds with Ricci curvature.
problem Understanding curvature decay in flat manifolds.
method Construct solutions to Yamabe flow and analyze curvature decay.
result If curvature decays quickly, manifold must be flat.
Gated attention improves model curvature, enhancing performance on nonlinear tasks.
problem Understanding the geometric implications of gating in attention mechanisms.
method Modeling attention outputs as Gaussian distributions and analyzing Fisher--Rao geometry.
result Gated attention enables non-flat geometries, including positively curved manifolds.
Open manifolds with nonnegative Ricci curvature have virtually abelian fundamental groups if they escape from bounded balls at a small rate.
problem Understanding the fundamental groups of open manifolds with nonnegative Ricci curvature.
method Analyzing the escape rate of minimal geodesic loops and relating it to the fundamental group's properties.
result If an open manifold has a small escape rate, its fundamental group is virtually abelian.
Improved gap for mean curvature of biharmonic hypersurfaces in spheres.
problem Improving bounds on mean curvature for biharmonic hypersurfaces.
method Analyzing complete CMC proper-biharmonic hypersurfaces in Euclidean spheres.
result Enhanced gap result for mean curvature range.
The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.
problem Proving lower bounds for Gaussian-weighted \(L^2\)-curvature integrals of self-shrinkers.
method Combining normal coordinate functions with weighted Poincaré inequalities and first-eigenvalue estimates.
result Explicit lower bounds in terms of entropy for closed self-shrinkers, leading to curvature gaps.
Estimates scalar curvature without nonnegativity, showing gap phenomenon on manifolds.
problem Estimating scalar curvature without curvature nonnegativity assumption.
method Derive estimates for scalar curvature and mean curvature on manifolds and domains.
result Show that metrics on even dimensional manifolds with nonzero Euler characteristic are ε-gap distance extremal.
We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…
The paper proves gap results for self-shrinkers in -mean curvature flow.
problem Understanding the gap in properties of self-shrinkers in -mean curvature flow.
method Proving gap results using a modified second fundamental form and a differential operator.
result Proper self-shrinkers are parabolic for a certain second-order differential operator.
Study shows fundamental gap of horoconvex domains in hyperbolic space has no positive lower bound.
problem Understanding the fundamental gap of horoconvex domains in hyperbolic space.
method Analysis of fundamental gap of geodesic balls as radius goes to infinity.
result Product of fundamental gap and square of diameter has no positive lower bound for horoconvex domains.
Estimates the mass gap for domains with integral Ricci curvature bounds.
problem Estimating the mass gap for domains with specific curvature conditions.
method Proving domains are John domains to estimate the first nonzero Neumann eigenvalue.
result Fundamental gap estimates for domains with integral Ricci curvature bounds.
In this sequel to [arXiv:1412.4114], we prove an energy gap result for Yang-Mills connections on principal -bundles, , over arbitrary, closed, Riemannian, smooth manifolds of dimension . We apply our version of the Lojasiewicz-Simon gradient inequality [arXiv:1409.1525, arXiv:1510.03815] to rem…