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
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48 results for “scalar curvature flow”
Study on the convergence rate of prescribed scalar curvature flow.
problem Prescribing scalar curvature on manifolds.
method Inspired by Yamabe flow convergence rate study, analyze the prescribed scalar curvature flow convergence rate.
result Determine the convergence rate of the prescribed scalar curvature flow.
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.
Study proves curvature flow existence on CR manifolds.
problem Existence of curvature flow solutions on CR manifolds.
method Prescribed Webster scalar curvature flow approach.
result Proves existence of curvature flow solutions on 3D CR manifolds.
In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.
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.
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.
3D metrics get scalar curvature bounds via IMCF.
problem Bounding scalar curvature for metrics.
method Inverse Mean Curvature Flow (IMCF) and Hawking mass monotonicity.
result Stability theorem for nonnegative scalar curvature.
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.
Calabi flow extended with bounded curvature integrals.
problem Extending Calabi flow on compact Kähler manifolds.
method Using bounded scalar curvature integrals.
result Calabi flow can be extended under certain curvature conditions.
Entire scalar curvature flow and hypersurfaces of constant scalar curvature in Minkowski spacemath.DG
We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacel…
Calabi flow works well with bounded curvature on compact manifolds.
problem Stability of extremal Kähler metrics under the Calabi flow.
method Extending the Calabi flow with scalar curvature bounds.
result Calabi flow converges exponentially to extremal Kähler metrics.
Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degreemath.DG
Unified flow approach to curvature problem on specific manifolds.
problem Prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree.
method Unified flow approach with conditions on curvature function .
result Flow converges to a conformal Hermitian metric with specified 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
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.
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 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.
In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function , which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges …
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…
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.
The study preserves lower bounds of total scalar curvature under specific metric convergence.
problem Preserving lower bounds of total scalar curvature on smooth manifolds.
method Used stability of Ricci flow and heat flow with Ricci flow background.
result Lower bound of weighted total scalar curvature is preserved under specified convergence conditions.
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.
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.
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 -curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduc…
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.
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 . We also establish a general form of the Hamilton-Tian Conjec…
The Yamabe flow on flat manifolds converges to a scalar flat metric.
problem Analyzing the convergence of Yamabe flow on asymptotically flat manifolds.
method Yamabe flow starting from an asymptotically flat manifold, convergence analysis.
result The flow converges to an asymptotically flat, scalar flat metric under certain conditions.
Let be a dimensional, closed Riemannian manifold of positive Yamabe invariant. For a smooth function on we consider a scalar curvature flow, that tends to prescribe as the scalar curvature of a metric conformal to . We show global existence and in case is not confo…
The paper studies a modified scalar curvature flow and proves convergence to a sphere.
problem Analyzing the convergence of a modified scalar curvature flow.
method Flow of starshaped hypersurfaces with a specific speed function, proving existence and convergence.
result The flow converges exponentially fast to a sphere, except for .
3-manifolds with positive scalar curvature and bounded geometry are contractible.
problem Characterizing 3-manifolds with positive scalar curvature and bounded geometry.
method Maximal weak solution to inverse mean curvature flow.
result Complete contractible 3-manifolds with positive scalar curvature and bounded geometry are .
Study pinches curvature under Laplacian G_2 flow, proving Weyl tensor norm blows up.
problem Pinching estimate on traceless Ricci curvature under Laplacian G_2 flow.
method Derive pinching estimate in terms of scalar curvature and Weyl tensor norm.
result Weyl tensor norm blows up at least at a certain rate under bounded scalar curvature.
The paper establishes bounds on scalar curvature on asymptotically flat manifolds.
problem Establishing scalar curvature bounds on asymptotically flat manifolds.
method Using Ricci-DeTurck flow and distributional scalar curvature, the paper derives bounds on scalar curvature.
result The scalar curvature lower bound under Ricci-DeTurck flow depends on the scalar curvature lower bound in the β-weak sense and time.
We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity.
Proves Ricci flow extensibility with integral norms.
problem Ricci flow singularities under integral norms.
method Bounded integral norms of curvature and scalar curvature.
result Extends Ricci flow under certain conditions.
New method removes scalar curvature assumption in Ricci flow smoothing.
problem Uniform bounds on scalar curvature and other factors for Ricci flow.
method Quantitative short-time existence of Ricci flow without scalar curvature assumption.
result Ricci flow smoothing for measure space limits, Gromov-Hausdorff compactness, and topological rigidity results.
New rigidity results for scalar curvature with stabilized conditions.
problem Establishing rigidity for scalar curvature with stabilized conditions.
method Construction of foliations and development of a monotone quantity using Ricci flow and heat equation.
result Generalized classical scalar curvature rigidity results to the \(T^{
times}\)-stabilized setting.
Proves rigidity of maps between manifolds with scalar curvature constraints.
problem Lipschitz rigidity problem in scalar curvature geometry.
method Harmonic map heat flow coupled with Ricci flow.
result Continuous maps between manifolds with scalar curvature constraints are either isometries or have scalar curvature strictly less than the sphere.
This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f. Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f.
Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in away from the singular set.
The pluriclosed flow preserves Vaisman condition on compact complex surfaces.
problem Preserving Vaisman condition under pluriclosed flow.
method Pluriclosed flow on compact complex surfaces.
result Preserves Vaisman condition if and only if starting metric has constant scalar curvature.
We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds o…
New proof for curvature and diameter estimates on Fano manifolds.
problem Curvature and diameter estimates for Kähler-Ricci flow on Fano manifolds.
method New Harnack estimate for special functions in space-time.
result Established new estimates for scalar curvature and diameter.
Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension
problem Convergence of scalar curvature in Kähler-Ricci flow
method Uniform -entropy or uniform Sobolev inequality
result Scalar curvature converges to negative Kodaira dimension
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.
This paper classifies complete self-shrinkers in R^(n+1) with nonnegative constant scalar curvature.
problem Classifying self-shrinkers with specific curvature conditions.
method Analyzing the mean curvature flow and using geometric properties.
result Complete classifications of n-dimensional self-shrinkers in R^(n+1) with nonnegative constant scalar curvature.
In this short note, we use classic computations for Kähler-Ricci flow to achieve scalar curvature bound for minimal manifold of general type.
Paper proves mass theorems for nonnegative scalar curvature metrics.
problem Proving mass theorems for metrics with nonnegative scalar curvature.
method New local inverse mean curvature flow with quantitative stability.
result Existence of isoperimetric sets in low regularity metrics.
We introduce a new geometric flow of Hermitian metrics which evolves an initial metric along the second derivative of the Chern scalar curvature. The flow depends on the choice of a background metric, it always reduces to a scalar equation and preserves some special classes of Hermitian structures, as balanced and Gaud…