Turing complete flow on 4-sphere preserves volume.
problem Creating a Turing complete flow on a 4-sphere.
method Smooth, conservative flow on the 4-sphere.
result Achieved a Turing complete, volume-preserving flow.
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48 results for “complete flow”
Ancient solutions to Kähler Ricci flow classified completely.
problem Ancient solutions to Kähler Ricci flow with nonnegative bisectional curvature.
method Complete classification of κ-noncollapsed, complete ancient solutions.
result Classification of all ancient solutions to Kähler Ricci flow with nonnegative bisectional curvature.
Flow on pseudoconvex domains without curvature bounds.
problem Existence and completeness of Kähler-Ricci flow.
method Established existence and completeness of Kähler-Ricci flow on pseudoconvex domains.
result Flow converges to complete Kähler-Einstein metric.
Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flowsmath.DG
Complete Ricci flow from singular 3D manifold with pseudolocality.
problem Constructing complete Ricci flow from singular 3D manifold.
method Combining pseudolocality results for singular and nonsingular flows.
result Ricci flow is complete for positive times under certain conditions.
Study proves short-term existence for harmonic maps under evolving metrics.
problem Analyzing harmonic maps under time-dependent metrics.
method Proves short-term existence for harmonic map heat flow coupled with a smooth family of complete metrics.
result Generalizes short-term existence results for harmonic map heat flow.
We prove uniqueness of instantaneously complete Ricci flows on surfaces. We do not require any bounds of any form on the curvature or its growth at infinity, nor on the metric or its growth (other than that implied by instantaneous completeness). Coupled with earlier work, particularly [23, 11], this completes the well…
Global existence of Yamabe flows on hyperbolic space proved without curvature bounds.
problem Global existence of Yamabe flows on hyperbolic space without completeness or curvature bounds.
method Instantaneously complete initial metrics, no curvature bounds required.
result Global existence of Yamabe flows on hyperbolic space of arbitrary dimension .
Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
problem Geodesic completeness and flow properties of compact Brinkmann spacetimes.
method Proof of geodesic completeness and flow properties of isotropic parallel vector fields in compact Brinkmann spaces.
result Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
Produces solutions to Kähler-Ricci flow with curvature bounds.
problem Finding complete bounded curvature solutions to Kähler-Ricci flow.
method Assumes smooth initial data uniformly equivalent to another complete bounded curvature metric. Extends to non-smooth and degenerate cases.
result Obtains existence time estimates and stability results for complex space forms.
Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.
problem Finding complete hyperbolic metrics on cusped 3-manifolds.
method Analogue of surface and compact 3-manifold flows, minimizing co-volume, extending through singularities.
result Existence of complete hyperbolic metric is equivalent to flow convergence.
In 2+1 dimensions, all complete spacetimes are cylindrical.
problem Understanding rigidity of Ricci flow spacetimes in dimensions.
method Analyzing complete and sufficiently regular spacetimes, showing they must be cylindrical.
result Every spatial slice is diffeomorphic to a fixed surface, and the spacetime is isometric to a classical Ricci flow.
The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to…
Willmore flow preserves low energy surfaces to planes.
problem Preserving low energy surfaces to planes under Willmore flow.
method Willmore flow equation for complete, properly immersed surfaces in Rn.
result Complete Willmore surfaces with low energy converge to planes.
We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures.
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
Uniform proof for Ricci flows on complete manifolds.
problem Proving short-time existence, uniqueness, and continuous dependence for Ricci flows.
method Using Koch-Lamm framework and tensor heat kernel estimates, with a new continuous dependence estimate.
result Uniform proof of short-time existence, uniqueness, and continuous dependence for Ricci flows.
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.
This paper classifies complete translating solitons in 3D space.
problem Understanding translating solitons for mean curvature flow.
method Full classification of complete translating graphs in R^3.
result A complete classification of complete translating graphs in R^3.
Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metr…
Second Ricci flow proves existence of Kaehler-Einstein metrics on noncompact manifolds.
problem Existence of Kaehler-Einstein metrics on noncompact Hermitian manifolds.
method Established short time existence and Shi's type estimate of second Ricci flow.
result Second Ricci flow proves existence of Kaehler-Einstein metrics on complete noncompact Hermitian manifolds.
Study geodesic flow on graph-related nilmanifolds, finding integrable and non-integrable cases.
problem Understanding geodesic flow on specific geometric structures.
method Construction of first integrals to show complete integrability.
result Examples of integrable and non-integrable geodesic flows.
This work completes Chern-Ricci flow on complex manifolds with incomplete data.
problem Existence and behavior of Chern-Ricci flows on complex manifolds.
method Analyzes the flow and potential flow on complex manifolds with incomplete initial data.
result Obtains existence results for Chern-Ricci flows and Kähler-Einstein metrics.
Study singularity formation in Ricci flow solutions.
problem Understanding singularity behavior in noncompact manifolds.
method Analyzing complete Ricci flow solutions.
result Evidence for stability of generalized cylinders as singularity models.
In this paper we survey the recent developments of the Ricci flows on complete noncompact Kähler manifolds and their applications in geometry.
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
Study on Yamabe flow on manifolds with singularities, proving removability.
problem Yamabe flow on manifolds with submanifold singularities.
method Analyzing the Yamabe flow on Riemannian manifolds of dimension minus a closed submanifold of dimension .
result Removability of singularities preserved along the Yamabe flow in certain cases.
The paper proves a Liouville theorem for heat flows on manifolds with specific curvature conditions.
problem Investigating heat flows on manifolds with specific curvature conditions.
method Gradient estimate and Liouville type theorem for ancient solutions.
result Established a Liouville theorem for -harmonic heat flows.
The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.
problem Exploring translating solitons in Riemannian products with powers of Gaussian curvature.
method Investigating -flows in Riemannian products for .
result Existence of complete rotational translating solitons for certain values of in .
We consider the flow on complete non-compact graphs. We prove that a complete graph evolves by the curvature up to some time depending on the radius of a sphere enclosed by the initial graph.
The study classifies complete Lagrangian self-shrinkers in 4D space.
problem Classifying complete Lagrangian self-shrinkers in 4D space.
method Complete classification of 2D complete Lagrangian self-shrinkers with constant squared norm of the second fundamental form.
result A complete classification for 2-dimensional complete Lagrangian self-shrinkers in with constant squared norm of the second fundamental form.
Study preserves planar and graphical properties of curves under elastic flow.
problem Maintaining planar and graphical properties of non-compact curves under elastic flow.
method Extended recent work on adapted elastic energy to derive thresholds for planar and graphical embeddedness.
result Derived new Li--Yau type inequality for complete planar curves.
Global existence and convergence of heat flow for p-harmonic maps.
problem Global existence and convergence of heat flow for p-harmonic maps between manifolds.
method Analysis of heat flow equations for p-harmonic maps.
result Global existence and convergence of heat flow for p-harmonic maps under certain conditions.
Paper proves uniqueness of Schrödinger flow on specific manifolds.
problem Proving uniqueness of Schrödinger flow on manifolds.
method Intrinsic proof using distance functions and gauge language.
result Uniqueness of Schrödinger flow from a general complete Riemannian manifold to a complete Kähler manifold.
The noncompact Yamabe flow can lead to incomplete metrics over infinite time.
problem Incompleteness of noncompact Yamabe flow solutions over infinite time.
method Analysis of long-time behavior of the noncompact Yamabe flow.
result Existence of a long-time solution that is complete for each time but converges to an incomplete metric.
We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually our proof also yields a (non-canonical) way to flow and regularize some interior region of a non-complete initial data sat…
We show that every complete entire self-shrinking solution on complex Euclidean space to the Kahler-Ricci flow must be generated from a quadratic potential.
The paper studies 4D Ricci flow manifolds with curvature constraints.
problem Investigating 4D Ricci flow manifolds with specific curvature conditions.
method Analyzing 4D manifolds with curvature constraints via Ricci flow.
result Proves topological and geometric gap theorems for maximal volume growth.
We describe a construction of complete embedded self-translating surfaces under mean curvature flow by desingularizing the intersection of a finite family of grim reapers in general position.
Study on ancient Ricci flows with positive curvature, proving noncollapsedness.
problem Characterizing ancient Ricci flows with positive sectional curvature.
method Analyzing complete and noncompact Type I ancient Ricci flows with positive sectional curvature.
result Ancient solutions are noncollapsed on all scales in complete and noncompact cases, and in even-dimensional closed cases.
Study on -hypersurfaces in weighted flow, focusing on volume and radius estimates.
problem Volume and radius estimates of -hypersurfaces in weighted flow.
method Volume comparison theorem and radius estimates analysis.
result Estimates for intrinsic diameter and extrinsic radius of -hypersurfaces.
The paper proves uniqueness of Ricci flow on noncompact manifolds.
problem Proving uniqueness of Ricci flow on noncompact manifolds.
method Pseudolocality theorem for -complete Ricci flow.
result Strong uniqueness theorem for -complete Ricci flow on Euclidean space.
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.
Proves uniqueness of Ricci flow with scaling invariant estimates.
problem Proving uniqueness of Ricci flow with scaling invariant curvature bound.
method Solving Ricci-harmonic map heat flow in unbounded curvature background.
result Complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique in dimension three.
Flow analysis leads to metric completion in Kähler geometry.
problem Analyzing Kähler-Ricci flows on compact manifolds.
method Normalized Kähler-Ricci flow convergence to Gromov-Hausdorff limits.
result Metric completion of twisted Kähler-Einstein metric.
Complete negative Kähler-Einstein metric found on Stein manifolds.
problem Existence of complete Kähler-Einstein metrics on Stein manifolds with negative curvature.
method Normalized Kähler-Ricci flow to deform metrics to complete negative Kähler-Einstein metric.
result Existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature.
Completing Prelaminationsmath.GT
Completes preliminary structures in 3D flows to foliations.
problem Characterizing completability of lamination pairs in 3-manifolds.
method General approach for various types of foliations and flows.
result Characterizes when lamination pairs can be completed to foliations.
New flow preserves singularities on incomplete manifolds.
problem Evolve incomplete manifolds with bounded curvature.
method Construct Ricci de Turck flow uniformly equivalent to initial metric.
result Any incomplete manifold can be evolved for a short time.
Gromov-Hausdorff convergence of time-slices of singular Ricci flows
problem Gromov-Hausdorff convergence of time-slices of singular Ricci flows
method Completion of singular Ricci flow with respect to a natural spacetime distance
result Gromov-Hausdorff convergence at the first singular time