Optimizes biharmonic map regularity using stratification methods.
problem Improving the known almost optimal regularity of biharmonic maps.
method Quantitative stratification method.
result Optimal regularity results for minimizing biharmonic maps.
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48 results for “Cheeger and Naber's result”
Singular sets on spaces with integral curvature bounds and diffeomorphism finiteness for manifoldsmath.DG
The paper improves estimates on singular sets in manifolds with integral curvature bounds.
problem Estimating the singular set of manifolds with integral curvature bounds.
method Using Gromov-Hausdorff limits and Cheeger-Naber methods.
result Improved Minkowski dimension estimate for singular sets.
Study curvature growth in 4D singularity models using Perelman's method.
problem Estimating curvature growth in 4D gradient Ricci soliton singularity models.
method Applied Perelman's point selection, Cheeger and Naber's fundamental result, and topological lemmas.
result Developed estimates for curvature growth in singularity models.
Extends space theory to singular spaces.
problem Analyzing Ricci flows of bounded curvature.
method Generalizes Cheeger, Colding, and Naber's theory.
result Theory applies to singular space limits of manifolds.
Unified proof of smooth fibration theorems for collapsed manifolds.
problem Smooth fibration theorems for collapsed manifolds with Ricci curvature bounded below.
method Generalized Reifenberg condition and transformation technique for almost splitting maps.
result Unified proof of smooth fibration theorems in many previous works.
Study controls curvature in Ricci flows using necks.
problem Controlling curvature in Ricci flows.
method Introducing necks of maximal symmetry and decomposing curvature into uniform bounds.
result Established -bounds on Riemann curvature tensor.
Study on energy-minimizing structures in complex geometry.
problem Existence and regularity of harmonic almost complex structures.
method Inspired by harmonic map theory, proving results similar to Schoen-Uhlenbeck and Cheeger-Naber.
result Proved existence and regularity similar to harmonic map theory.
In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper bound for the Euler characterisic, and a lower bound for the gradient of the potential at large distances. In this note, we show that the last two assumptions in fact …
3D Ricci flows have bounded diameter before Type I singularities.
problem Bounding the diameter of 3D Ricci flows before Type I singularities.
method Introduced a neck-region concept and proved packing measure Ahlfors regularity.
result Uniformly bounded diameter up to Type I singular time.
The paper extends regularity for -minimizing maps using a Reifenberg Theorem.
problem Quantitative regularity of -minimizing maps between Riemannian manifolds.
method Stratification of singular points based on almost-symmetries, followed by application of a Reifenberg-type Theorem.
result Upper bound on the Minkowski content of the singular set, and -rectifiability of the singular set.
Paper proves ε-regularity for shrinking Ricci solitons and Ricci flows.
problem Proving ε-regularity for critical metrics and Ricci flows.
method Constructing counterexamples and proving ε-regularity theorems.
result Proves ε-regularity for 4-dimensional shrinking Ricci solitons and partially confirms for Ricci flows.
Integral estimates for Ricci flows up to singular time.
problem Integral estimates for curvature tensor in Type I Ricci flows.
method Adapted quantitative stratification technique.
result Partial extension of curvature estimates to higher dimensions.
Convexity proven in Ricci shrinker limit spaces.
problem Understanding the structure of Ricci shrinker limits.
method Regular-singular decomposition and parabolic smoothing of distance functions.
result The regular part of any Ricci shrinker limit space is convex.
The paper bounds the shortest closed geodesic length in 4D manifolds.
problem Finding the shortest closed geodesic in 4D manifolds with specific curvature and volume constraints.
method Utilizes recent theorems on diffeomorphism finiteness by J. Cheeger and A. Naber.
result The length of a shortest closed geodesic is bounded by a function that depends on volume and diameter .
Let $\cM$ be a Brakke flow of -dimensional surfaces in . The singular set $\cS\subset\cM$ has a stratification $\cS^0\subset\cS^1\subset...\cS$, where $X\in \cS^j$ if no tangent flow at has more than symmetries. Here, we define quantitative singular strata $\cS^j_{η,r}$ satisfying $\cup_{η>0}\cap_{0<r} …
Extends positive mass theorem to arbitrary dimensions using a new inductive scheme.
problem Overcoming singularities in the Schoen-Yau proof for arbitrary dimensions.
method Inductive scheme combining shielding principle, conformal blow-up, and Cheeger-Naber bound.
result Proof of positive mass theorem in arbitrary dimensions.
Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded belowmath.DG
Study confirms conjectures on Ricci limit spaces and their topological properties.
problem Understanding the topological structure of noncollapsed Ricci limit spaces.
method Analysis of tangent cones and application of manifold recognition theorems.
result Cross-sections of tangent cones at points in 4D spaces are homeomorphic to a fixed spherical space form.
The paper constructs manifolds with infinite holes from a given manifold.
problem Creating manifolds with infinite holes from a given manifold.
method Constructing a sequence of -dimensional manifolds with that approximate the original manifold and have an infinite number of connected components.
result The constructed manifolds have dense boundary with an infinite number of connected components and no open subset topologically a manifold.
Paper reconciles different Ricci flow approaches and proves weak solutions.
problem Proving weak solutions for Ricci flows with singularities.
method Introducing a novel hitting estimate for Brownian motion, compensating for lack of lower heat kernel bounds.
result Every noncollapsed limit of Ricci flows and singular Ricci flows are weak solutions.
Study of splitting maps in Type I Ricci flows for understanding singular set structure.
problem Understanding the structure of the singular set in non-collapsed Ricci limit spaces.
method Construction and investigation of almost splitting maps on Ricci flows that are almost self-similar.
result Sharp splitting maps remain splitting maps at smaller scales under certain conditions.
We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.
Simplified proof for Cheeger's isoperimetric constant.
problem Cheeger's isoperimetric constant
method Simplified proof of Buser's result
result Simplified proof for Cheeger's isoperimetric constant
New upper bound for Cheeger constant of hyperbolic surfaces.
problem Bounding the Cheeger constant of hyperbolic surfaces.
method Random construction based on Poisson--Voronoi tessellation.
result The Cheeger constant of closed hyperbolic surfaces is less than that of the hyperbolic plane.
The paper proves inequalities for Steklov eigenvalues on finite graphs.
problem Eigenvalues of Laplacians for reversible Markov chains and Steklov eigenvalues.
method Generalized Cheeger inequalities, convergence results, and resolvent convergence.
result Sharp estimate for the first non-trivial Steklov eigenvalue.
Lower bounds for eigenvalues on manifolds with boundary conditions.
problem Eigenvalue bounds for manifolds with boundary conditions.
method Proving lower bounds for the first non-trivial eigenvalue using Cheeger-type constants.
result Results in the spirit of Cheeger's inequality for manifolds with boundary conditions.
We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L2-de Rham cohomology theory satisfying Poincare duality. We prove that this cohomology …
Cheeger inequalities defined for graph limits, proving key inequalities.
problem Defining and proving Cheeger inequalities for graph limits.
method Introducing graphon and graphing concepts, proving inequalities.
result Proved Cheeger and Buser inequalities for graphons and graphings.
This paper standardizes Cheeger deformations on fiber bundles with compact groups.
problem Developing a unified approach to Cheeger deformations on fiber bundles.
method Systematic introduction and re-proving existing results using Cheeger deformations.
result Unified approach to Cheeger deformations on fiber bundles with compact structure groups.
Sharp Gaussian isoperimetry proven along Ricci flow.
problem Proving sharp Gaussian isoperimetric inequality for Ricci flow.
method Using monotonicity formula to prove inequality.
result Exact Gaussian enlargement theorem and concentration estimates.
The paper bounds Cheeger ratios of eigenfunctions and their level sets.
problem Understanding geometric features of Riemannian manifolds through eigenfunctions.
method Constructive upper bounds on Cheeger constants using eigenvalues and eigenfunctions.
result Upper bounds on Cheeger ratios of eigenfunction level sets and their superlevel sets.
Extends Cheeger's method to Lie groupoid actions on manifolds.
problem Smooth Lie group actions on manifolds with singularities.
method Extension of Cheeger's deformation techniques to Lie groupoid actions.
result Explicit sectional curvature description of the deformation.
Paper extends variational formula for Bismut-Cheeger eta form, proving key theorems in K-theory.
problem Extending variational formula for Bismut-Cheeger eta form without kernel bundle assumption.
method Twisting spin Dirac operators by isomorphic vector bundles, proving -graded additivity.
result Analytic index in differential K-theory is a well-defined group homomorphism, and Riemann-Roch-Grothendieck theorem in K-theory.
This is a survey article with a limited list of references (as required by the publisher) which appears in the Encyclopedia of Mathematical Physics, eds. J.-P. Francoise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006. vol.4, pp.94--104.
Investigates Cheeger sets in rotationally invariant domains and their free boundaries.
problem Properties of Cheeger sets in rotationally invariant domains.
method Analyzes properties of Cheeger sets and their free boundaries, using Delaunay surfaces and constant mean curvature.
result For convex domains, free boundaries consist of spheres and nodoids; for nonconvex domains, unduloids or cylinders can also appear.
We introduce certain relative differential characters which we call Cheeger-Chern-Simons characters. These combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as the Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, the Cheeger-Chern-Simons …
The Cheeger constant increases under Ricci flow on spheres.
problem Behavior of the Cheeger constant under Ricci flow.
method Evolution identities for parallel curves and viscosity formulation of .
result The Cheeger constant is non-decreasing under Ricci flow on surfaces diffeomorphic to .
From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data cloudsmath.SP
Study shows convergence rates for Cheeger cuts on data clouds.
problem Optimizing graph cuts for clustering data sampled from a manifold.
method Analyzes statistical properties of Cheeger cuts on proximity graphs built from data.
result Obtains high probability convergence rates for Cheeger constant and cuts.
The study extends a theorem to bundles on manifolds with boundaries.
problem Extending a theorem to bundles on manifolds with boundaries.
method Using recent anomaly results by Brüning, Ma, and Zhang, the theorem is generalized for a general flat bundle with a unimodular restriction to the boundary.
result An analogous statement for a general flat bundle is proven.
Study on Cheeger constant in specific hyperbolic 3-manifolds.
problem Analyzing Cheeger constant in convex co-compact hyperbolic 3-manifolds.
method Examining Cheeger constant as a functional in the space of specific hyperbolic 3-manifolds.
result Global maximum of Cheeger constant is uniquely attained at the Fuchsian locus.
New inequality for special forms on manifolds.
problem Understanding coexact 1-forms on manifolds.
method Proved a Cheeger-like inequality.
result New inequality for coexact 1-forms.
The paper studies eigenvalues and Cheeger constants on symmetric graphs.
problem Characterizing eigenvalues and Cheeger constants on symmetric graphs.
method Characterization of the first eigenfunction via sign condition, and calculation of Cheeger constants using the limit of -Laplacian eigenvalues.
result Identifies Cheeger constants of symmetric graphs and their quotients.
Develops axiomatic framework for differential cohomology and constructs generalized Cheeger-Simons characters.
problem Differential cohomology in the relative case.
method Axiomatic framework and construction of generalized Cheeger-Simons characters.
result Definition of integration map for fibre with boundary.
Study calculates Cheeger constants and small eigenvalues of Maass cusp forms.
problem Understanding small eigenvalues of Maass cusp forms on hyperbolic surfaces.
method Computed Cheeger constants of hyperbolic surfaces and searched for small eigenvalues of Maass cusp forms numerically.
result Found evidence and computed small eigenvalues of Maass cusp forms.
Random hyperbolic surfaces have low Cheeger constants.
problem Estimating Cheeger constants of random hyperbolic surfaces.
method Modeling random hyperbolic surfaces using ideal triangles and analyzing their Cheeger constants.
result Generic hyperbolic surfaces have Cheeger constants less than 3/2π + ε.
Extends Cheeger-Colding Theory to conic Kahler-Einstein metrics.
problem Proving Yau-Tian-Donaldson conjecture for Fano varieties with specific singularities.
method Extension of Cheeger-Colding Theory.
result Technical tool for proving Y-T-D conjecture.
Proof of Reifenberg theorem in metric spaces, expanding on Cheeger and Colding's work.
problem Proving the Reifenberg theorem in metric spaces using Gromov-Hausdorff distance.
method Detailed proof of Cheeger and Colding's result, expanding on their arguments.
result BiLipschitz version of the Reifenberg theorem in metric spaces.
Study on simply connected manifolds with discrete isometric actions and bounded quotient diameter.
problem Characterizing simply connected manifolds with discrete isometric cocompact group actions.
method Analyzing sequences of manifolds with bounded diameter and Ricci curvature lower bound, using Gromov-Hausdorff convergence and Lie group theory.
result The quotient space of the limit manifold is simply connected, and the fundamental group is generated by loops in the maximal torus orbit.
In this paper, we extend Su-Zhang's Cheeger-Mueller type theorem for symmetric bilinear torsions to manifolds with boundary in the case that the Riemannian metric and the non-degenerate symmetric bilinear form are of product structure near the boundary. Our result also extends Bruening-Ma's Cheeger-Mueller type theorem…