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.
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.
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 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.
The paper proves a gap theorem for Kähler manifolds with specific curvature properties.
problem Proving a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature.
method Using advanced geometric analysis techniques to prove the gap theorem.
result Generalizes earlier results on Kähler manifolds with these curvature properties.
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.
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.
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.
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.
The paper proves gap results for self-shrinkers in r-mean curvature flow.
problem Understanding the gap in properties of self-shrinkers in r-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.
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 …
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.
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.
The paper proves gap properties for critical metrics under specific conditions.
problem Proving gap properties for critical metrics under divergence-free Bach tensor condition.
method Analyzing critical point equation of total scalar curvature with divergence-free Bach tensor.
result Proves gap properties for n≥5 and a similar condition for n=4. 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.
Sharp curvature estimates for expanding Ricci solitons in various dimensions.
problem Estimating curvature bounds for expanding Ricci solitons.
method Sharp lower and upper bounds derived for scalar curvature under specific conditions.
result Sharp curvature estimates provided for expanding Ricci solitons in dimensions three and four.
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…
Extends curvature gap characterization for minimal surfaces in a ball.
problem Characterization of minimal surfaces in a ball.
method Pinching condition on second fundamental form.
result Extension to higher codimension.
New gaps found in metric curvature.
problem Negative curvature metrics with separated length spectra.
method Topology-based separation of length spectra.
result Exponential gaps in length spectra for negatively curved metrics.
New gap theorem for hypersurfaces with constant mean curvature in space forms.
problem Finding a gap for hypersurfaces with constant mean curvature and scalar curvature.
method Analyzing the relationship between the squared length of the second fundamental form and the mean curvature.
result Proving a new gap theorem for hypersurfaces with constant mean curvature and constant scalar curvature.
Formulae quantify gaps in geodesic quadrilaterals on manifolds.
problem Infinitesimal gaps in geodesic quadrilaterals on manifolds.
method Formulas for curvature and torsion based on quadrilateral gaps.
result Torsion and curvature can be uniquely recovered from quadrilateral gaps.
New estimates show spectral gap stability in RCD spaces, close to Beta distribution.
problem Stability of spectral gap bounds in metric-measure spaces.
method Combines L1-functional inequality and Stein's method. result Sharp quantitative estimate for spectral gap stability.
New curvature measure defined for graphs, with bounds on diameter and spectral gap.
problem Defining curvature for graphs and proving its properties.
method Solving linear systems to compute curvature; applying minimax theorem.
result Graphs with positive curvature have bounded diameter and spectral gap.
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 gap results for constant mean curvature surfaces in Euclidean and hyperbolic spaces.
problem Understanding the properties of constant mean curvature surfaces.
method Proving natural inequalities and using them to show that CMC surfaces in Euclidean and hyperbolic spaces are either spheres, cylinders, or disks.
result Gap theorems for CMC surfaces in Euclidean and hyperbolic spaces complete the picture.
The study finds energy gaps for Yang-Mills fields on Kähler surfaces.
problem Finding energy gaps for Yang-Mills fields on Kähler surfaces.
method Proving an L2 energy gap result for Yang-Mills connections on Kähler surfaces with positive scalar curvature. result Proves energy gap results for Yang-Mills fields on Kähler surfaces.
The study proves a gap theorem for shrinking gradient Ricci solitons with specific curvature and volume conditions.
problem Characterizing shrinking gradient Ricci solitons with given curvature and volume constraints.
method Combining Günther's volume comparison theorem and Yokota's gap theorem, the study proves a gap theorem.
result Complete shrinking gradient Ricci solitons with specific curvature and volume constraints are isometric to the Gaussian soliton.
Formula found for surfaces in Sol_3, leading to gap results.
problem Finding constant mean curvature surfaces in Sol_3.
method Computed Laplacian of squared norm of second fundamental form, used Simons type formula.
result Gap results for compact constant mean curvature surfaces.
In this short note, we find a new gap phenomena on Riemannian manifolds, which says that for any complete noncompact Riemannian manifold with nonnegative Ricci curvature, if the scalar curvature decays faster than quadratically, then it is Ricci flat.
Study pinching constants for Kähler manifolds with positive curvature.
problem Pinching constants of Kähler manifolds with positive holomorphic sectional curvature.
method Apply techniques from Riemannian pinching theory to Kähler geometry.
result Prove a gap theorem for Kähler manifolds with almost quarter-pinched holomorphic sectional curvature.
This paper extends gap theorems for submanifolds in hyperbolic space.
problem Understanding gap phenomena for submanifolds in hyperbolic space.
method Generalizes existing results using Simons' formula and eigenvalue estimates.
result Proves a gap theorem for hypersurfaces with constant scalar curvature in hyperbolic space.
Lu conjecture proven for minimal 2-spheres and surfaces under certain conditions.
problem Discreteness of constant scalar curvatures of compact minimal submanifolds in unit spheres.
method Refined Simons' first gap theorem and Yau's theorems for high-codimensional submanifolds.
result Lu's conjecture for minimal 2-spheres and surfaces proved under inequality conditions.
The paper proves a gap theorem and entropy noncollapsing for ancient solutions to the Ricci flow.
problem Understanding ancient solutions to the Ricci flow and their properties.
method Analyzing asymptotic entropy and proving gap theorems and noncollapsing conditions.
result Finite asymptotic entropy implies kappa-noncollapsing on all scales for complete ancient solutions with nonnegative curvature operator.
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.
In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with ∣A∣2≤1 in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.
Paper studies a new curvature system and proves rigidity and gap theorems.
problem Extending CPE conjecture to manifolds with specific structures.
method Introduces (φ−CPE) system and proves rigidity and gap theorems. result Proves rigidity and gap theorems for (φ−CPE) solutions. Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.
The study provides energy estimates for Willmore surfaces and derives a gap statement.
problem Analyzing the tracefree curvature of Willmore surfaces.
method Proves ε-regularity result for tracefree curvature with bounded second fundamental form.
result Derives a gap statement for surfaces of the specified type.
Study rigidity of spectral gap on Finsler manifolds with specific curvature bounds.
problem Rigidity of spectral gap on Finsler manifolds with Ricci curvature bound.
method Analysis of spectral gap, splitting phenomena, and needle decomposition.
result Rigidity results for spectral gap, logarithmic Sobolev, and Bakry-Ledoux inequalities.
Study surfaces with parallel mean curvature in spheres, proving rigidity results.
problem Characterize surfaces with parallel mean curvature in unit spheres.
method Establish Simons-type integral identities and apply to rigidity and gap estimates.
result Obtain first two sharp endpoint gaps and rigidity estimates.
Researchers prove rigidity for spectral gap on special metric spaces.
problem Proving rigidity for spectral gap on RCD(K,∞)-spaces. method Lift of eigenfunctions to Wasserstein space, theory of regular Lagrangian flows.
result Sharp spectral gap achieved only by splitting off a 1-dimensional Gaussian space.
The paper studies 4D Ricci flow manifolds with curvature constraints.
problem Investigating 4D Ricci flow manifolds with specific curvature conditions.
method Analyzing 4D manifolds with curvature constraints via Ricci flow.
result Proves topological and geometric gap theorems for maximal volume growth.
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.
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 L∞--curvature gap of complex vector bundles on a compact manif…
In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time T of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …
Flat space for manifolds with tiny curvature.
problem Understanding manifolds with curvature concentration.
method Analyzing non-compact manifolds with non-negative Ricci curvature and small curvature concentration.
result Manifolds with curvature concentration are flat.
Local gaps in Ricci shrinkers depend only on dimension.
problem Understanding local properties of Ricci shrinkers.
method Proved local versions of Ricci curvature and entropy gap theorems.
result Local gaps depend only on dimension, not global entropy.
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…