Compactness theorem for manifolds with boundary proved.
problem Proving convergence of manifolds with boundary.
method Reduction to Hamilton's compactness theorem for manifolds without boundary.
result Cheeger-Gromov convergence for a subsequence of manifolds with bounded geometry.
The article proves finiteness results for 2D convex hypersurfaces using surgery on mean curvature flow.
problem Proving finiteness for 2D convex hypersurfaces in Rn+1. method Using mean curvature flow with surgery for 2 convex hypersurfaces.
result Proves extrinsic finiteness results in the spirit of Cheeger's compactness theorem.
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.
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 …
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.
Generalizes Candel's theorem on curvature of laminated surfaces.
problem Finding curvature of laminated surfaces given certain conditions.
method Proves a generalized theorem using elliptic PDEs and Cheeger-Gromov topology.
result Unique laminated metric exists for given curvature function.
Defines intrinsically Hölder sections in metric spaces.
problem Characterizing Hölder sections in metric spaces.
method Introducing intrinsically Hölder graphs, proving compactness, regularity, and extension theorems.
result Establishes properties for intrinsically Hölder graphs, including vector space, convex set, and equivalence relation.
Study four-dimensional Ricci flow using branching curves.
problem Characterize Type I singularities in four-dimensional Ricci flow.
method Associate one-parameter families of curves to points in the product of two projective lines.
result Characterize singularity models in four-dimensional Ricci flow.
We prove the equality of the L2-analytic torsion and the intersection R torsion of the even dimensional finite metric cone over an odd dimensional compact manifold.
The paper proves a compactness theorem for spaces with Bakry-Emery Ricci tensor.
problem Compactness in spaces with Bakry-Emery Ricci tensor.
method Proving f-mean curvature comparison and defining a Myers-type compactness theorem. result Improves a result from Soylu by using a weaker condition on f′(t). Sharp spectral theorem splits certain non-compact manifolds.
problem Proving spectral splitting for non-compact manifolds with specific curvature conditions.
method Sharp spectral analysis and geometric splitting theorem.
result Non-compact manifolds split as RimesN under given curvature constraints. We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form CD(0,1). Even though we have to allow warping in our splitting, we are able to recov…
The Cheeger-Gromoll theorem is adapted for groups, revealing geometric properties of virtually abelian groups.
problem Understanding the curvature of groups and its relation to geometric properties.
method Developed a new curvature notion for groups and proved a splitting theorem.
result Geometric characterization of virtually abelian groups.
Compactness theorem for Riemannian manifolds with volume and curvature bounds.
problem Investigating the regularity of limit spaces of Riemannian manifolds.
method Local volume growth condition, compactness theorem, different convergence notion.
result Compactness theorem for Riemannian manifolds with Lp curvature bounds and volume growth assumption. 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.
Extends Bonnet-Myers theorem with new generalizations.
problem Generalizing Bonnet-Myers theorem.
method Complementary generalization of existing extensions by Calabi and Cheeger-Gromov-Taylor.
result New theorem extending previous extensions of Bonnet-Myers.
Compactness results for Hermitian manifolds help understand Type IIB flow.
problem Understanding singularities in the Type IIB flow.
method Formulated convergence criteria for Hermitian manifolds and applied them to the Type IIB flow.
result Existence of singularity models for the Type IIB flow.
Study extends compactness theorems to weighted manifolds with integral curvature bounds.
problem Estimating diameter of weighted manifolds under curvature constraints.
method Extended Sprouse's compactness theorems to weighted manifolds with integral curvature bounds. Used ε-range to handle specific cases. Extended segment inequality to weighted manifolds.
result Proved theorems for weighted manifolds with effective dimension ≤ 1 and ≥ dimension.
Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.
problem Investigating properties of Kähler manifolds with specific curvature conditions.
method Conformal perturbation method.
result Established structure and splitting theorems for Kähler manifolds with non-negative mixed curvature.
The paper compares two torsion invariants in complex vector bundles.
problem Comparing two torsion invariants in complex vector bundles.
method Constructing Bismut-Lott analytic torsion classes and showing they coincide with Igusa-Klein torsions.
result Bismut-Lott analytic torsion classes coincide with Igusa-Klein torsions for trivial flat line bundles.
By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology …
Study Cheeger inequalities for Riemannian manifolds with boundary.
problem Estimating Steklov eigenvalues on Riemannian manifolds with boundary.
method Establish Cheeger-type inequalities using isocapacitary constants.
result Cheeger inequalities for Steklov eigenvalues on compact and non-compact manifolds.
In this paper, we mainly establish a Cheeger type finiteness theorem for Berwald manifolds. In order to do this, we study the injectivity radius and the convex radius of a Finsler manifold. A Cheeger type estimate on injectivity radii for Finsler manifolds is given and the existence of the center of mass of a Berwald m…
Generalizes tools for studying collapsed manifolds to new geometry.
problem Studying collapsed manifolds with bounded sectional curvature.
method Generalizes fibration and stability theorems for compact group actions on manifolds with local bounded Ricci covering geometry.
result Two generalized results used in Xiaochun Rong's work on almost flat manifolds.
New insights into possible Gauss curvatures on S2.
problem Understanding possible Gauss curvatures of metrics on S2. method Employing the smooth Cheeger-Gromov compactness theorem to obtain a priori estimates and proving a proper Fredholm map with well-defined degree.
result Existence and non-existence results for Gauss curvatures in stable regions.
In this paper, we prove two generalized versions of the Cheeger-Gromoll splitting theorem via the non-negativity of the Bakry-Émery Ricci curavture on complete Riemannian manifolds.
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…
We prove equality between the renormalized Ray-Singer analytic torsion and the intersection R-torsion on a Witt-manifold with cusps, up to an error term determined explicitly by the Betti numbers of the cross section of the cusp and the intersection R-torsion of a model cone. In the first step of the proof we compute e…
In this paper, we study the volume growth property of a non-compact complete Riemannian manifold X. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to study gradient Ricci solitons. We also show that on X, for any q∈(0,∞),…
The paper extends a splitting theorem for a specific type of tensor in Riemannian geometry.
problem The extension of a splitting theorem for a new type of tensor in Riemannian geometry.
method The approach involves extending the spectral Cheeger-Gromoll splitting theorem to smooth metric measure spaces.
result The theorem allows for the isometric splitting of a manifold under certain conditions on the tensor and its eigenvalues.
Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p in M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff…
The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.
problem Extending the Bakry-Émery-Ricci tensor and proving comparison theorems.
method Generalizations of the drifted Laplacian and Bakry-Émery-Ricci tensor, mean curvature comparison theorem, Myers-type theorem, Cheeger-Gromoll splitting theorem.
result Proved a version of the mean curvature comparison theorem and its consequences.
The paper proves a theorem about torsion and Morse functions for covering spaces.
problem Proving a theorem about torsion and Morse functions for covering spaces.
method Using a fiberwise Morse function and the Novikov-Shubin invariant.
result Proves the Cheeger-Müller theorem for L2-analytic torsion form. 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.
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 spinc Dirac operators by isomorphic vector bundles, proving Z2-graded additivity. result Analytic index in differential K-theory is a well-defined group homomorphism, and Riemann-Roch-Grothendieck theorem in R/Z K-theory. 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.
We study the analytic torsion of the cone over an orientable odd dimensional compact connected Riemannian manifold W. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a fu…
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.
This is a short survey of Cheeger and Kleiner's nonembeddability theorem for Heisenberg group into L1.
We will show a theorem of a type of Cheeger and Muller for a noncompact complete hyperbolic threefold of finite vulume. As an application we will compute a special value of Ruelle L-function at the origin for a unitary local system which is cuspidal.
Maps from curved spaces have a minimum energy bound.
problem Finding the minimum energy of maps between curved spaces.
method Analyzing the energy of maps from manifolds with nonnegative Ricci curvature to other manifolds.
result The energy of maps is bounded by a constant determined by the target's geometry, with equality for totally geodesic maps.
A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature and uniformly controlled injectivity radius, and extract a subsequence that conv…
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 logh. result The Cheeger constant is non-decreasing under Ricci flow on surfaces diffeomorphic to S2. When a discrete group admits a convex-cocompact action on a non-compact rank-one symmetric space, there is a natural lower bound for the Hausdorff dimension of the limit set, given by the Ahlfors regular conformal dimension of the boundary of the group. We show that equality is achieved precisely when the group stabili…
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.
In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion and demonstrate that Bismut-Zhang's equivariant Cheeger-Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and …
Study harmonic flow of Spin(7)-structures on compact 8-manifolds.
problem Isometric flow of Spin(7)-structures on compact 8-manifolds.
method Establishing Shi-type estimates, self-similar solutions, monotonicity formula, compactness theorems, and Bryant-type description.
result Conditions for long-time existence and characterisation of singularities.
Study heat kernel and spectrum on manifolds with wedge singularities.
problem Analyzing the spectrum and heat kernel on manifolds with wedge singularities.
method Uniform constructions of the resolvent and heat kernel on manifolds with corners, using spectral gap and analytic torsion.
result Obtain a Cheeger-Müller theorem relating analytic torsion and Reidemeister torsion.