New scalar curvature defined from Ollivier-Ricci curvature for graphs.
problem Defining scalar curvature for graphs and point clouds.
method Defining a new scalar version of Ollivier-Ricci curvature and proving its convergence.
result The new scalar curvature converges to scalar curvature for sampled manifolds.
Survey paper analyzes curvature estimates for 4D gradient Ricci solitons.
problem Analyzing curvature estimates for different types of 4D gradient Ricci solitons.
method Comparison and new estimates provided for 4D gradient steady Ricci solitons.
result Sharp curvature estimate ∣Rm∣≤CR for gradient steady Ricci solitons with positive Ricci curvature. Defines projective Ricci curvature and proves rigidity for sprays.
problem Defining and studying projective Ricci curvature.
method Introduced projective Ricci-flat sprays and studied Randers metrics.
result Global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature.
Paper classifies Randers metrics based on Ricci curvature properties.
problem Investigating isotropic projective Ricci curvature in Randers metrics.
method Classification of Randers metrics based on isotropic projective Ricci curvature properties.
result Randers metric of isotropic projective Ricci curvature is reversible if and only if it is of square projective Ricci curvature.
Equivalence proven between different Ricci curvature definitions.
problem Equivalence of various Ricci curvature definitions.
method Proof of equivalence between different Ricci curvature definitions.
result Equivalence proven between several Ricci curvature definitions.
Defined Ricci curvature on simplicial complexes and proved bounds.
problem No specific problem stated; generalization of graph Ricci curvature to simplicial complexes.
method Modified Ricci curvature definition for simplicial complexes and proved bounds.
result Upper and lower bounds of Ricci curvature on simplicial complexes.
We introduce some new curvature quantities such as conformal Ricci curvature and bi-Ricci curvature and extend the classical Myers theorem under these new curvature conditions. Moreover, we are able to obtain the Myers type theorem for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature. Some top…
Proves metrics with positive intermediate Ricci curvature on complex manifolds.
problem Establishing metrics with positive intermediate Ricci curvature on complex manifolds.
method Canonical variation and surgery techniques.
result Existence of metrics with positive intermediate Ricci curvature on various examples.
New discrete Ricci curvature for directed networks developed.
problem Directed networks require a new curvature measure.
method Extended Forman-Ricci curvature for directed networks, incorporating vertex and edge weights, and edge direction.
result New curvature measure captures higher-order correlations in directed networks.
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 et/2. Study classifies Ricci solitons with specific curvature conditions.
problem Classifying gradient shrinking Ricci solitons with nonnegative orthogonal bisectional curvature.
method Proves classification results under invariant conditions without curvature bounds.
result Obtains new results on ancient solutions for Ricci and Kähler-Ricci flows.
Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.
problem Understanding the asymptotic geometry of ancient Ricci flows with nonnegative Ricci curvature.
method Analyze tangent flows at infinity and use estimates for noncollapsed F-limit metric solitons.
result Two dichotomy theorems for ancient Ricci flows: either the asymptotic volume ratio is zero or every tangent flow is a Ricci flat cone.
The paper extends Ricci curvature in Finsler geometry and finds conditions for specific metrics.
problem Characterizing and finding conditions for specific metrics in Finsler geometry.
method Introducing weighted projective Ricci curvature and analyzing Randers and Kropina metrics.
result Conditions for metrics to have weighted projective Ricci flat curvature.
Combinatorial approach to α-Ricci and Lin-Lu-Yau Ricci curvatures on graphs
problem Curvature formulas for α-Ricci and Lin-Lu-Yau Ricci curvatures on graphs method Combinatorial construction of optimal transport plans and exact formulas
result Combinatorial proof of known curvature formulas
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.
New curvature defined via graph resistances leads to Ricci flow.
problem Defining curvature on graph edges for analysis.
method Introducing Ricci--Foster curvature based on effective resistances and studying Ricci flow.
result Existence of solutions to Ricci flow on short time intervals, preservation of nonnegative curvature.
Two Ricci curvature discretizations correlate in complex networks.
problem Comparing two Ricci curvature definitions for complex networks.
method Empirical comparison of Forman-Ricci and Ollivier-Ricci curvatures.
result Forman-Ricci curvature correlates highly with Ollivier-Ricci curvature in real-world networks.
The paper classifies certain types of Ricci solitons with specific curvature properties.
problem Classifying gradient steady Ricci solitons with linear curvature decay.
method Analyzing the curvature properties and using classification techniques.
result Classification of 3D and 4D steady Ricci solitons with nonnegative curvature under linear decay.
Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.
problem Proving Hamilton's pinching conjecture for three-manifolds.
method Ricci flow with scale-invariant curvature decay and pinching preservation.
result Hamilton's pinching conjecture is proven without additional hypotheses.
Introduces new curvature concept for Kähler manifolds.
problem Optimizing curvature constraints for projective Kähler manifolds.
method Introduces weighted orthogonal Ricci curvature and proves vanishing theorems.
result Proves optimal curvature constraints for projective Kähler manifolds.
We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using which we prove the short-time existence of Ricci flow solutions o…
The paper explores curvatures on graphs and their implications for Ricci flatness.
problem Comparing and understanding different curvature notions on graphs and their implications for Ricci flatness.
method Analyzing Ollivier Ricci curvature and Bakry-Émery curvature on combinatorial graphs, investigating graph products, and proving curvature properties.
result Non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-Émery curvature under specific conditions.
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.
Positive Ricci curvature achieved by adding edges in a graph.
problem Achieving positive Ricci curvature in graphs.
method Adding edges to a complete graph to increase Ricci curvature.
result Least number of edges needed for positive Ricci curvature.
The study finds bounds on Ricci curvature for possibly collapsed spaces.
problem Bounding Ricci curvature in collapsed spaces.
method Lower bound on Ricci curvature, volume collapse consideration.
result Existence of bounds on local Lp norm of Ricci curvature. The paper studies weighted Ricci curvatures and characterizes Randers metrics.
problem Characterizing Randers metrics with weighted Ricci curvatures.
method General weighted Ricci curvatures and characterization of Randers metrics.
result Characterization of Randers metrics with almost isotropic weighted Ricci curvatures.
Study on curvature decay in steady Ricci solitons, proving dichotomy.
problem Curvature decay in steady Ricci solitons.
method Established a dichotomy for curvature decay in specific types of solitons.
result Proved a dichotomy on curvature decay for certain steady Ricci solitons.
Study on a weaker curvature condition for Kähler manifolds.
problem Understanding Kähler manifolds with nonpositive k-Ricci curvature. method Introducing almost nonpositive k-Ricci curvature and analyzing the twisted Kähler-Ricci flow. result Compact Kähler manifolds with almost nonpositive k-Ricci curvature have nef canonical line bundles. Paper bounds local curvature for Ricci-harmonic flow under Ricci bounded condition.
problem Local curvature estimates for Ricci-harmonic flow.
method Explicit bound of Δg(t)u(t) and local curvature estimates.
result Local curvature estimates for generalized Ricci flow and conjecture for Einstein's scalar field equations.
If π:M→B is a Riemannian Submersion and M has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that B must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of…
We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all n-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with ∣Ric∣≤K must satisfy $|Rm|\…
The paper examines Ricci solitons with convex potential and finds them flat and split.
problem Characterizing Ricci solitons with specific properties.
method Analyzes the Ricci curvature and potential function of Ricci solitons.
result Gradient Ricci solitons with convex potential are Ricci flat and isometrically split.
Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.
problem Rigidity of Kähler manifolds with nonnegative Ricci curvature.
method Analysis of Kähler manifolds with specific properties.
result Complete noncompact Kähler surface with nonnegative Ricci curvature, Euclidean volume growth, and quadratic curvature decay is biholomorphic to the resolution of an affine algebraic variety.
The paper extends topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.
problem Generalizing topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.
method Study of the Splitting Theorem and geodesic loops to infinity property in relation to spaces with nonnegative N-Bakry Émery Ricci curvature.
result If Mn is a complete, noncompact Riemannian manifold with nonnegative N-Bakry Émery Ricci curvature where N>n, then Hn−1(M,Z) is 0. Solving Ricci curvature problem on homogeneous spaces.
problem Finding metrics with prescribed Ricci curvature on homogeneous spaces.
method Surveying recent progress in solving the problem for homogeneous spaces.
result Recent progress in solving the problem for homogeneous spaces.
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.
In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient Kähler-Ricci solitons with positively pinched Ricci curvature sho…
New curvature condition proves rigidity of Bryant Ricci solitons.
problem Proving rigidity of Bryant Ricci solitons.
method Introducing a new curvature-pinching condition and proving rigidity results.
result Rotationally symmetric solutions of steady Ricci solitons are rigid under the new curvature condition.
Study on stable minimal hypersurfaces under Ricci curvature constraints.
problem Stability of weighted minimal hypersurfaces under Ricci curvature bounds.
method Derive geometric consequences and prove a Schoen-Yau type criterion.
result Structure theorem for three-dimensional weighted manifolds of non-negative Ricci curvature.
Extends Bochner's theorem to include small positive Ricci curvature.
problem Classical Bochner theorem limitations.
method Extensions with small positive Ricci curvature.
result Isometry group is finite for small positive curvature.
The Ricci flow preserves positivity on Stiefel manifolds.
problem Preserving positivity of Ricci curvature on Stiefel manifolds.
method Normalized Ricci flow on Stiefel manifolds.
result Normalized Ricci flow evolves metrics with mixed Ricci curvature into positive ones.
Study on gradient steady Kähler Ricci solitons with specific curvature properties.
problem Characterize gradient steady Kähler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature.
method Analyzing the structure of these solitons and their universal covering spaces.
result Gradient steady Kähler Ricci solitons are quotients of specific manifolds.
Develops structure theory for Ricci shrinkers without curvature restrictions.
problem Understanding the structure of Ricci shrinkers without curvature conditions.
method Structure theory development for non-collapsed Ricci shrinkers.
result Curvature estimates of Ricci shrinkers based on non-collapsing constant.
The study examines Ricci solitons and curvature inheritance on Robinson-Trautman spacetimes.
problem Investigating Ricci solitons and curvature inheritance in Robinson-Trautman spacetimes.
method Analyzing the existence of Ricci solitons and curvature inheritance properties on Robinson-Trautman spacetimes.
result Robinson-Trautman spacetimes admit various types of Ricci solitons and curvature inheritance.
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.
Smooth 3D flows from non-smooth starting points.
problem Creating smooth Ricci flows from non-smooth initial conditions.
method Generalized singular Ricci flow applied to 3D complete manifolds.
result Existence of smooth Ricci flows starting from non-smooth initial conditions.
Study on modified Ricci curvature on graphs, proving rigidity and deriving formulas.
problem Understanding Ricci curvature on graphs, especially for specific graph types.
method Introduced modified Ricci curvature, established rigidity theorem, derived formulas for strongly regular graphs.
result Rigidity theorem for complete graphs and explicit formulas for strongly regular graphs.
In 2D, synthetic curvature notions match.
problem Matching synthetic curvature notions in 2D.
method Synthetic curvature notions in 2D.
result Synthetic curvature notions match in 2D.