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.
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48 results for “Ricci curvature bounds”
Study compares isoperimetric profiles on manifolds with integral Ricci curvature bounds.
problem Comparing isoperimetric profiles on manifolds with integral Ricci curvature bounds.
method Extending previous work, the study uses integral bounds on Ricci curvature to prove comparison results for isoperimetric profile functions.
result Comparison results for the Isoperimetric profile function in manifolds with integral bounds on Ricci curvature.
The paper provides estimates for higher-order Ricci curvature along Kähler-Ricci flows.
problem Estimating higher-order curvature along Kähler-Ricci flows on compact Kähler manifolds.
method Proving uniform bounds for Ricci curvature and scalar curvature in various orders and norms.
result A geometric obstruction causes a specific third-order derivative of Ricci curvature to blow up at rate .
Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global -regularity for 4D Ricci flow with integral scalar curvature bound.
Uniform entropy bound for Ricci shrinkers with bounded curvature.
problem Bounding entropy for Ricci shrinkers with specific curvature constraints.
method Establishing uniform entropy bounds for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
result Uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds.
Lower bounds for geodesic ball volume in 3D with Ricci curvature constraints.
problem Finding volume bounds in 3D manifolds with Ricci curvature limits.
method Providing lower bounds for geodesic ball volume with upper bounds on Ricci curvature.
result Established lower bounds for geodesic ball volume under Ricci curvature constraints.
Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.
problem Collapsing geometry of Riemannian manifolds with Ricci curvature constraints.
method Locally bounded Ricci covering geometry and Ricci flow smoothing techniques.
result Volume collapsed Calabi-Yau manifolds admit Ricci-flat Kähler metrics and compatible Killing structures.
Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for we show the existence of bounds on the local norm of the Ricci curvature that depend only on the dimension and which improve with volume collapse.
Convergence of Ricci-limit spaces under bounded Ricci curvature and local covering geometry Imath.DG
The study extends convergence theorems for Ricci-limit spaces with bounded curvature.
problem Understanding convergence properties of Ricci-limit spaces with bounded curvature.
method Establishing -regularities and applying Fukaya's fibration theorem.
result Optimal generalization of Fukaya's fibration theorem to limit spaces.
Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.
problem Extending Ricci flow theory under Kato-type curvature bounds.
method Extends non-collapsed Ricci flow existence theory to Kato-type lower bounds.
result Compact three-dimensional non-collapsed strong Kato limit spaces are homeomorphic to smooth manifolds.
We use a first-order energy quantity to prove a strengthened statement of uniqueness for the Ricci flow. One consequence of this statement is that if a complete solution on a noncompact manifold has uniformly bounded Ricci curvature, then its sectional curvature will remain bounded for a short time if it is bounded ini…
Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvaturesmath.DG
The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.
problem Understanding the behavior of parabolic frequency under Ricci flow and Ricci-harmonic flow.
method Investigates the monotonicity of parabolic frequency for solutions of linear and heat equations with bounded curvatures.
result Establishes monotonicity results for parabolic frequency under specific curvature conditions.
The paper proves inequalities under Bakry-Émery-Ricci curvature bounds.
problem Proving functional inequalities under lower Bakry-Émery-Ricci curvature bounds.
method Lower -Bakry-Émery-Ricci curvature bounds with -range.
result Proves Cheng type inequality and local Sobolev inequality.
Study shows Gromov's Betti number bound fails for certain intermediate Ricci curvatures.
problem Gromov's Betti number bound for sectional curvature bounded below does not hold for intermediate Ricci curvatures.
method Established a surgery result for Riemannian metrics with and showed failure of Gromov's bound for specific ranges of .
result Gromov's Betti number bound fails for when .
Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.
problem Quantifying unique continuation for Schrödinger operators on manifolds with specific curvature conditions.
method Proving quantitative unique continuation estimates for Schrödinger operators on manifolds with Ricci curvature bounded below.
result Upper bound for energy range and constant in terms of Ricci curvature and parameters of relatively dense set.
The study establishes conditions for orientability in spaces with lower Ricci curvature bounds.
problem Conditions for orientability in spaces with lower Ricci curvature bounds.
method Equivalent characterizations of orientability using Ricci limit and RCD spaces.
result Four-manifolds with Ricci curvature bounded below and volume non-collapsing are uniformly locally orientable.
Eigenvalue estimates for Beltrami-Laplacian under specific curvature conditions.
problem Estimating eigenvalues of the Beltrami-Laplacian on manifolds with Bakry-Émery Ricci curvature.
method Using Bakry-Émery Ricci curvature conditions, the paper derives lower bounds for eigenvalues of the Beltrami-Laplacian.
result Lower bounds for eigenvalues of the Beltrami-Laplacian depend on curvature, gradient bounds, dimension, and diameter of the manifold.
The paper proves bounded curvature for Kähler Ricci shrinker surfaces.
problem Understanding the curvature of Kähler Ricci shrinker surfaces.
method Proved bounded sectional curvature using earlier work.
result Complete classification of all Kähler Ricci shrinker surfaces.
The Ricci flow preserves product structures with instantaneous curvature bounds.
problem Preserving product structures under Ricci flow with curvature constraints.
method Proving a constant ε exists such that if a solution splits as a product at time 0 and has bounded curvature, it splits for all time.
result A constant ε exists depending on dimension such that if a solution splits as a product at time 0 and has curvature bounded by ε/t, it splits for all time.
Uniform bounds for eigenvalues of Hodge Laplacian on manifolds with lower Ricci curvature.
problem Establishing bounds for eigenvalues of Hodge Laplacian under lower Ricci curvature.
method Using geometric assumptions including lower Ricci curvature, injectivity radius, and diameter bounds.
result Uniform eigenvalue bounds for the Hodge Laplacian and connection Laplacian.
The paper proves properties of minimal graphs on manifolds with Ricci curvature bounds.
problem Understanding properties of minimal graphs on manifolds with Ricci curvature constraints.
method Gradient estimates and Ahlfors-Khas'minskii duality in nonlinear potential theory.
result Positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant, and complete, parabolic manifolds with Ricci curvature bounds have the half-space property.
Proves optimal volume growth for certain nonnegative Ricci curvature manifolds.
problem Volume growth of manifolds with nonnegative Ricci curvature and positive bi-Ricci curvature.
method Analyzes bi-Ricci curvature and uses Gromov's volume bound conjecture analogy.
result Proves optimal volume growth for manifolds with specific curvature conditions.
There are two primary goals to this paper. In the first part of the paper we study smooth metric measure spaces (M^n,g,e^{-f}dv_g) and give several ways of characterizing bounds -Kg\leq \Ric+\nabla^2f\leq Kg on the Ricci curvature of the manifold. In particular, we see how bounded Ricci curvature on M controls the anal…
Proves equivalence of two types of Ricci curvature bounds.
problem Equivalence of distributional and synthetic Ricci curvature bounds.
method Analyzes weighted Riemannian manifolds with specific smoothness conditions.
result Proves equivalence of Ricci curvature bounds under given conditions.
The paper proves scalar curvature lower bounds along Ricci flow on compact manifolds.
problem Preserving scalar curvature lower bounds along Ricci flow.
method Analyzing Ricci flow on compact manifolds with initial metrics having scalar curvature lower bounds.
result If initial metric has scalar curvature lower bound, it is preserved along Ricci flow.
Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.
problem Understanding singularity types in long-time solutions of Chern-Ricci flow.
method Extended results from Kähler-Ricci flow to Chern-Ricci flow, focusing on uniform bounds on torsion and curvature.
result Uniform bounds on torsion and curvature for solutions starting from metrics of the same class.
Ancient Ricci flows with nonnegative curvature operator have bounded entropy.
problem Conditions for bounded entropy in ancient Ricci flows.
method Used Perelman's entropy and Hamilton's trace Harnack inequality.
result Curvature operator nonnegativity is not necessary for bounded entropy.
Ricci flows with bounded scalar curvature cannot develop Type I singular points.
problem Ricci flows with bounded scalar curvature
method Local singularity analysis
result Scalar curvature must blow up at a Type I rate at each Type I point
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
Defining Pointwise Lower Scalar Curvature Bounds for Metrics with Regularization by Ricci Flowmath.DG
Surveying Ricci flow for weak lower scalar curvature bounds.
problem Creating local definitions for weak lower scalar curvature bounds for metrics.
method Using Ricci flow to define and analyze weak lower scalar curvature bounds.
result Properties and applications of Ricci flow in defining weak lower scalar curvature bounds.
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.
Abstract reviews known and open questions on spaces with lower Ricci bounds.
problem Understanding the structure and regularity of spaces with lower Ricci curvature bounds.
method Review of known results and presentation of open questions.
result Presentation of new open questions in the field.
The paper improves curvature estimates for Ricci flow solutions with bounded scalar curvature.
problem Proving curvature estimates for Ricci flow solutions with bounded scalar curvature.
method Localised weighted curvature integral estimates for solutions to Ricci flow.
result Integral curvature estimates imply a uniform bound on the spatial norm of the Riemannian curvature tensor.
We analyze an upper bound on the curvature of a Riemannian manifold, using "root-Ricci" curvature, which is in between a sectional curvature bound and a Ricci curvature bound. (A special case of root-Ricci curvature was previously discovered by Osserman and Sarnak for a different but related purpose.) We prove that our…
We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we ob…
New decay estimates for scalar curvature of steady gradient Ricci solitons.
problem Understanding scalar curvature behavior in steady gradient Ricci solitons.
method Using μ-bubbles introduced by Gromov.
result Provide new decay estimates for scalar curvatures.
We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-Émery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvat…
For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.
We prove a pseudolocality type theorem for compact Ricci Flow under local integral bounds of curvature. The main tool is Local Ricci Flow introduced by Deane Yang in [4] and Pseudolocality Theorem of Perelman in [3]. We also study L^p bounds for the derivatives of curvature and smooth extension of Local Ricci Flow.
The paper studies steady solitons with curvature decay and proves their smoothness.
problem Analyzing the properties of steady solitons with curvature decay.
method Bootstrap regularity in harmonic coordinates using the soliton equation.
result Steady gradient Ricci solitons are asymptotically cylindrical under certain curvature decay conditions.
The paper studies Ricci flow with specific curvature and volume constraints, proving convergence and curvature bounds.
problem Analyzing Ricci flow with Ricci curvature and volume constraints.
method Proving convergence and curvature bounds for Ricci flow with specific constraints.
result Ricci flow with specified constraints converges to a flat cone or static flow.
We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times \in[0, T), then the curvature tensor has to be uniformly bounded as well.
The paper shows inequality and rigidity for manifolds with integral Ricci curvature.
problem Analyzing structures of manifolds with integral Ricci curvature.
method Using segment inequality and similar methods as in \cite{CC1}, derive almost rigidity structure results.
result Sharp Hölder continuity result holds in the limit space of manifolds with integral Ricci curvature bound.
Study compares synthetic and distributional Ricci curvature bounds.
problem Comparing synthetic and distributional approaches to lower Ricci curvature bounds.
method Analyzes synthetic via weak displacement convexity and distributional via non-negativity of Ricci-tensor.
result Distributional bounds imply entropy bounds for metrics and vice versa for under convergence condition.
Optimal volume limit found for Kähler manifolds with positive Ricci curvature.
problem Bounding the volume of Kähler manifolds with positive Ricci curvature.
method Using δ-invariants and Newton--Okounkov bodies.
result Derive the optimal volume upper bound and new characterization of the complex projective space.
Estimates the first eigenvalue for embedded hypersurfaces in manifolds with Ricci curvature bounds.
problem Estimating the first eigenvalue for embedded hypersurfaces in manifolds with Ricci curvature bounds.
method Establishes a lower bound for the first nonzero eigenvalue of the Laplacian on embedded hypersurfaces in a closed oriented Riemannian manifold under Ricci curvature and sectional curvature bounds.
result The estimate depends on ambient curvature bounds, normal injectivity radius, and geometry of the hypersurface.
Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.
problem Bounding curvature of regularized metrics with constraints on Ricci tensor and injectivity radius.
method Mollification of riemannian metrics, uniform -harmonic radius bounds, Ricci tensor bounds, injectivity radius bounds.
result Uniform estimate on the change of sectional curvature for regularized metrics.