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.
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 ε-regularity for 4D Ricci flow with integral scalar curvature bound. Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced hh-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduc…
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.
We use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which are all satisfied if the manifold is curvature homogeneous. This leads to a comp…
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
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.
In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scala…
New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
This paper sets lower bounds for scalar curvatures in Ricci flow singularity models.
problem Understanding scalar curvatures in Ricci flow singularity models.
method Developed high-dimensional theory of Hamilton's Ricci flow, including new monotonicity formulas, compactness theorem, and partial regularity theory.
result Obtained a quadratic decay lower bound for the scalar curvature in 4-dimensional non-Ricci-flat steady soliton singularity models.
Surveying Ricci flow for weak lower scalar curvature bounds.
problem Creating local definitions for weak lower scalar curvature bounds for C0 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.
The paper characterizes solitons and estimates scalar curvature.
problem Characterizing and estimating scalar curvature of generalized Ricci-Yamabe solitons.
method Characterization and estimation of scalar curvature through soliton properties.
result Conditions for scalar curvature to be constant and estimation of Ricci curvature.
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
5D shrinking Ricci solitons with constant scalar curvature are rigid.
problem Characterizing 5D shrinking gradient Ricci solitons with constant scalar curvature.
method Proving rigidity by showing they are finite quotients of a known space.
result 5D shrinking gradient Ricci solitons with constant scalar curvature are rigid.
Enhances Ricci flow theorem with scalar curvature bound.
problem Improving no-local-collapsing theorem of Ricci flow.
method Derives improved theorem under scalar curvature bound condition.
result Refines Perelman's no-local-collapsing theorem.
Study on positive scalar curvature and its impact on Ricci limit spaces.
problem The influence of uniformly positive scalar curvature on Ricci limit spaces.
method Investigates uniformly positive scalar curvature on non-collapsed Ricci limit spaces.
result Proves a limit space splits at most n-2 lines or R-factors.
In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of…
Paper proves rigidity for Ricci solitons with specific conditions.
problem Understanding the properties of Ricci solitons under various conditions.
method Analyzes shrinking and expanding Ricci solitons with specific constraints.
result Compact shrinking Ricci solitons are Einstein if the potential function is controlled.
New metrics with non-negative scalar curvature are always Ricci-flat on certain surgeries.
problem Understanding metrics with non-negative scalar curvature on surgeries of manifolds.
method Analyzing spin surgeries and their impact on metrics with non-negative scalar curvature.
result Complete metrics with non-negative scalar curvature are Ricci-flat on certain surgeries.
The paper defines weak lower scalar curvature bounds for C0 metrics and shows their stability under Ricci flow.
problem Defining and proving stability of weak lower scalar curvature bounds for metrics with low regularity.
method Proposes local definitions of weak lower scalar curvature bounds for C0 metrics, shows stability under perturbation, and defines a Ricci flow for C0 initial data. result Weak lower scalar curvature bounds are preserved under Ricci flow from C0 initial data. Study explores how scalar functionals evolve under Ricci flow.
problem Understanding the evolution of functionals involving scalar quantities under Ricci flow.
method Deriving explicit expressions for the time derivative of integrals of scalar functionals under extended Ricci flow.
result Explicit expressions for the time derivative of integrals involving scalar functionals under Ricci flow.
Paper proves a Liouville theorem for solitons with constant curvature.
problem Understanding harmonic functions on specific geometric structures.
method Proved a Liouville theorem without gradient estimates.
result Finite dimensionality of harmonic functions with polynomial growth.
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.
New findings on gradient expanding Ricci solitons with finite scalar curvature ratio.
problem Understanding the behavior of gradient expanding Ricci solitons with finite scalar curvature ratio.
method Analyzing complete gradient expanding Ricci solitons with nonnegative Ricci curvature.
result Riemann curvature tensor must have at least sub-quadratic decay for finite asymptotic scalar curvature ratio.
The paper examines conditions for Ricci solitons to be Ricci flat or Einstein.
problem Conditions for Ricci solitons to be Ricci flat or Einstein.
method Study of evolution equations of scalar and Ricci curvatures under Ricci flow.
result Conditions for Ricci solitons to be Ricci flat or Einstein.
Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
Alternative proof for 4D shrinking Ricci solitons with constant scalar curvature.
problem Proving the structure of four-dimensional shrinking gradient Ricci solitons with constant scalar curvature.
method Analyzing the asymptotic geometry at infinity.
result Alternative proof that such solitons are finite quotients of R^2 x S^2.
Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.
problem Defining scalar curvature for non-smooth spaces and flows.
method Gaussian integral approach to scalar curvature, applied to manifolds and flows.
result Characterizes Ricci flows as minimal super-Ricci flows.
The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.
problem Understanding the rigidity properties of Sasaki-Ricci solitons as singularity models.
method Established fundamental equations and criteria for transverse rigidity, proving key results about scalar curvature and Weyl tensor.
result Low-dimensional Sasaki-Ricci solitons with constant scalar curvature are Sasaki-Einstein, and those with harmonic Weyl tensor are finite quotients of the sphere.
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
The abstract conjectures and proves conditions for scalar-flat Kähler surfaces with specific tensor properties.
problem Conditions for scalar-flat Kähler surfaces with special tensor properties.
method Conjecture and prove in three special cases.
result The conjecture is proven in three special cases.
Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension
problem Convergence of scalar curvature in Kähler-Ricci flow
method Uniform μ-entropy or uniform Sobolev inequality result Scalar curvature converges to negative Kodaira dimension
The paper studies curvature properties under a specific type of flow on spaces with conical singularities.
problem Preserving curvature properties (Ricci curvature and scalar curvature) under a flow with conical singularities.
method Ricci de Turck flow, preserving conical structure, additional assumptions for scalar curvature positivity.
result Positivity of scalar curvature is preserved under the flow with additional assumptions.
We show that the scalar curvature of a steady gradient Ricci soliton satisfying that the ratio between the square norm of the Ricci tensor and the square of the scalar curvature is bounded by one half, is boundend from below by the hyperbolic secant of one half the distance function from a fixed point.
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.
We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity.
The paper proves pseudolocality theorems for Ricci flows on incomplete manifolds.
problem Pseudolocality of Ricci flows on incomplete manifolds.
method Proves pseudolocality theorems for Ricci flows under specific curvature and isoperimetric conditions.
result Constructs solutions of Ricci flow in balls with pseudolocality property.
Study finite time singularities in Ricci flow with bounded scalar curvature.
problem Understanding finite time singularities in Ricci flow with bounded scalar curvature.
method Analyzing blow-up sequences of locally Type I singularities.
result Every blow-up sequence of a locally Type I singularity has a specific property.
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 L2 norm of the Riemannian curvature tensor. This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions g…
In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension ≥4. We also establish a general form of the Hamilton-Tian Conjec…
The paper extends Ricci flow theory with Type-I scalar curvature bounds, proving entropy convergence and characterizing singular sets.
problem Extending Ricci flow theory with Type-I scalar curvature bounds.
method Type-I rescaling procedure and entropy analysis of conjugate heat kernels.
result Entropy of Ricci flow solutions converges to soliton entropy, characterizing singular sets.
The study classifies gradient Ricci solitons with specific vector fields.
problem Characterizing gradient Ricci solitons with closed conformal vector fields.
method Analyzing properties of gradient Ricci solitons with constant scalar curvature and closed conformal vector fields.
result Gradient Ricci solitons with these properties are isometric to specific spaces.
The paper proves conditions under which critical point metrics are Einstein.
problem Proving that critical point metrics are Einstein under specific curvature constraints.
method Analyzing the traceless Ricci operator and scalar curvature constraints.
result The conjecture that critical point metrics are Einstein is proven under certain curvature conditions.
Sharp bound on scalar curvature integral in 3-manifolds.
problem Bounding the integral of scalar curvature on 3-manifolds.
method Geodesic ball analysis with nonnegative Ricci curvature.
result Integral of scalar curvature is bounded by 8πR for large radii. In this very short note we prove a lower bound for the scalar curvature of certain steady gradient Ricci solitons.
The study proves the rigidity of certain gradient Ricci solitons with constant scalar curvature.
problem Proving the rigidity of specific gradient Ricci solitons with constant scalar curvature.
method Analyzing the properties of gradient shrinking Ricci solitons with constant scalar curvature and nonnegative Ricci curvature.
result The study proves that these solitons are isometric to finite quotients of specific spaces.
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.