In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cro…
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48 results for “cross curvature flow”
Long Time Existence of the Cross Curvature Flow in 3-Manifolds with Negative Sectional Curvaturemath.DG
The cross curvature flow preserves negative sectional curvature in 3-manifolds for all time.
problem Preserving negative sectional curvature in 3-manifolds under the cross curvature flow.
method Maximum principle to prove long-time existence of the flow.
result The flow exists for all time and converges to a hyperbolic metric.
Chow and Hamilton introduced the cross curvature flow on closed 3-manifolds with negative or positive sectional curvature. In this paper, we study the negative cross curvature flow in the case of locally homogenous metrics on 3-manifolds. In each case, we describe the long time behavior of the solutions of the correspo…
Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross curvature flowsmath.DG
In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the…
We show that there exists a suitable neighborhood of a constant curvature hyperbolic metric such that, for all initial data in this neighborhood, the corresponding solution to a normalized cross curvature flow exists for all time and converges to a hyperbolic metric. We show that the same technique proves an analogous …
Estimates for hypersurfaces in CROSSes show cylindrical profiles near singularities.
problem Analyzing the mean curvature flow of hypersurfaces in complex or quaternionic projective spaces.
method Proving apriori estimates on principal curvatures under a pinching assumption.
result The asymptotic profile near a singularity is either strictly convex or cylindrical.
Examines properties of negatively curved 3-manifolds and their embeddings, introduces Cross Curvature Flow.
problem Understanding negatively curved three-manifolds and their embeddings.
method Reviews Cross Curvature Flow as a tool, examines rigidity properties, reviews embeddings into Minkowski space.
result Fixed Einstein volume solutions are integrable solutions, answering a question posed by Chow and Hamilton.
In this paper, we use the DeTurck trick to study the short-time existence of solutions to the Dirichlet and Newmann boundary problems of the cross curvature flow on 3-manifolds with boundary.
Recently, B.Chow and R.S.Hamilton introduced the cross curvature flow on 3-manifolds. In this paper, we analyze two interesting examples for this new flow. One is on a square torus bundle over a circle, and the other is on a bundle over a circle. We show that the global flow exist in both cases. But on the form…
Two ancient solutions to Gauss curvature flow are identified for cylinders.
problem Classifying ancient solutions to Gauss curvature flow in cylinders.
method Assumption of cylinder cross-section bounded convexity, analysis of asymptotic behavior.
result Only two ancient solutions identified: translating soliton and compact oval solution.
Geodesic flows on surfaces have specific fractional-linear integrals related to constant cross-ratios.
problem Characterizing geodesic flows on surfaces with fractional-linear integrals.
method Proving the dimension of fractional-linear integrals and giving a geometric criterion.
result The dimension of fractional-linear integrals is either 3 or 5, corresponding to constant curvature.
Study shows how to preserve Lagrangian condition in mean curvature flow on Kim-McCann metrics.
problem Preserving Lagrangian condition in mean curvature flow on Kim-McCann metrics.
method Expressed mean curvature flow within generalized mean curvature flow framework.
result Lagrangian condition is preserved along the flow.
The classic 2pi-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2pi-me…
This paper is concerned with properties of maximal solutions of the Ricci and cross curvature flows on locally homogeneous three-manifolds of type SL(2,R). We prove that, generically, a maximal solution originates at a sub-Riemannian geometry of Heisenberg type. This solves a problem left open in earlier work by two of…
Total variation and mean curvature flows on a Lie group quotient enhance and denoise crossing structures.
problem Preserving crossing curvilinear structures in image enhancement and denoising.
method Lifting images to the homogeneous space , applying PDEs for TVF and MCF, and using locally optimal differential frames.
result Better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings compared to data-driven diffusions.
We flow a hypersurface in Euclidean space by mean curvature flow with a Neumann boundary condition, where the boundary manifold is any torus of revolution. If we impose the conditions that the initial manifold is compatible and does not contain the rotational vector field in its tangent space, then mean curvature flow …
Flow of curved surfaces converges to a specific shape over time.
problem Behavior of curved surfaces over time.
method Addressed through the -Gauss curvature flow for .
result Flow converges to a translating soliton determined by initial conditions.
We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional curvature, we establish short time existence and a pair of monotonicity formulas for solutions to the flow. One of these formulas…
We demonstrate that the uniqueness of solutions to a broad class of parabolic geometric evolution equations can be proven via a direct and essentially classical energy argument which avoids the DeTurck trick entirely. Previously, we have used a variation of this technique to give an alternative proof and slight extensi…
Ricci flows with almost maximal extinction time are nearly round.
problem Understanding the rigidity of Ricci flows with positive curvature.
method Analyzing the relationship between Ricci flows, their extinction times, and curvature properties.
result Ricci flows with almost maximal extinction time are nearly round.
Markov partitions for the geodesic flow on compact Riemann surfaces of constant negative curvaturemath.DG
The paper constructs Markov partitions for geodesic flow on hyperbolic surfaces.
problem Understanding Markov partitions for general hyperbolic flows.
method Rigorous construction of Markov partitions for geodesic flow on Riemann surfaces of constant negative curvature.
result Explicit forms of rectangles and local cross sections provided for the geodesic flow.
The study of Kähler manifolds and their curvature properties.
problem Characterizing Kähler manifolds using cross quadratic bisectional curvature.
method Analyzing properties of Kähler manifolds with CQB and CQB, using the Kähler-Ricci flow.
result Compact Kähler manifolds with positive CQB or CQB are Fano, and nonnegative CQB or CQB leads to a Fano manifold if the universal cover does not contain a flat de Rham factor.
The paper discusses mean curvature flow with surgeries for 2-convex hypersurfaces, focusing on neck detection and gluing.
problem Mean curvature flow with surgeries for 2-convex hypersurfaces.
method Establishing neck detection, gluing cross sections, and using harmonic spherical parametrisation.
result Uniqueness, existence, and overlapping properties for normal parametrisations on -cylindrical hypersurface necks.
Nonnegative cross-curvature in infinite dimensions: synthetic definition and spaces of measuresmath.MG
Proposes a variational NNCC formulation for infinite dimensions.
problem Optimization and gradient flows in infinite-dimensional settings.
method Variational formulation of NNCC on c-convex domains.
result Wasserstein spaces inherit NNCC from their base space.
New perspective on Ricci flow on spheres using Minkowski spacetime.
problem Classifying singularity models for null mean curvature flow in Minkowski spacetime.
method Equivalence of 2d-Ricci flow and null mean curvature flow on lightcones.
result Classification of singularity models for null mean curvature flow.
In this paper, we study the backward Ricci flow on locally homogeneous 3-manifolds. We describe the long time behavior and show that, typically and after a proper re-scaling, there is convergence to a sub-Riemannian geometry. A similar behavior was observed by the authors in the case of the cross curvature flow.
Study Ricci flow on spaces with conical singularities, proving existence and curvature estimates.
problem Analyzing Ricci flow on spaces with conical singularities.
method Existence proof for Ricci flow, curvature estimates, and tangent flow analysis.
result Existence of a solution to Ricci flow for a specific class of spaces.
We give a simple, direct proof of the backward uniqueness of solutions to a class of second-order geometric evolution equations including the Ricci and cross-curvature flows. The proof, based on a classical argument of Agmon-Nirenberg, uses the logarithmic convexity of a certain energy quantity in the place of Carleman…
Smooth conic Kähler metrics with uniform curvature bound.
problem Constructing smooth conic Kähler metrics with bounded curvature.
method Using upper bisectional curvature bound and choosing a new background metric.
result Constructing a smoothing sequence for conic metrics.
The Yamabe flow preserves conical singularities under certain conditions.
problem Preserving conical singularities under the Yamabe flow.
method Using maximal -regularity theory for conically degenerate operators.
result Asymptotic expansion of the evolving metric near conical tips.
The paper connects complex normalizing flows to Kähler-Ricci flows using geometric and statistical perspectives.
problem Understanding the relationship between complex normalizing flows and Kähler-Ricci flows.
method Develops connections between complex normalizing flows and Kähler-Ricci flows by relating the log determinant to Ricci curvature and using a Bayesian perspective.
result Reconciles the complex normalizing flow and Kähler-Ricci flow, showing they are related under certain conditions.
Study classifies algebraic solitons on 3D Lie groups.
problem Classifying solitons on Lie groups.
method Investigated cross curvature flow and second order renormalization group flow.
result Complete classification of left invariant algebraic solitons.
Norms on curves on surfaces classify Birkhoff cross sections.
problem Classifying Birkhoff cross sections on surfaces.
method Defining norms on homology groups and interpreting integer points.
result Integer points in dual unit balls classify isotopy classes of Birkhoff cross sections.
Proves estimate similar to De Lellis-Müller on Minkowski lightcone.
problem Estimating spacelike cross sections of the Minkowski lightcone.
method Geometric scaling invariant estimate, using singularity models and almost-Schur lemma.
result Spacelike cross sections are -close to a round surface.
This paper proves geodesic curvature measures are bounded for curves near cross cap singularities.
problem Boundedness of geodesic curvature measures near cross cap singularities.
method Analyzes intrinsic cross cap singularities and extends Gauss-Bonnet formula.
result Proves boundedness of geodesic curvature measures for curves near cross cap singularities.
Study shows integrating OFI from multiple levels improves price impact explanation but not forecasting.
problem Explaining and forecasting price movements in equity markets using OFI.
method Systematic approach to combine OFIs from multiple levels into an integrated variable, testing multi-asset models with and without cross-impact terms.
result Lagged cross-asset OFIs improve future return forecasting but not contemporaneous price impact.
Study curvature and torsion from cross-ratios in discrete curves.
problem Define curvature and torsion for discrete curves using cross-ratios.
method Use Möbius invariant point-insertion-rule to construct circles and express torsion using cross-ratio.
result Discrete curvature and torsion defined using cross-ratios converge to smooth curvature and torsion as sampling density increases.
The paper solves problems related to curvature on a 3-sphere.
problem Prescribing positive cross curvature on the three-dimensional sphere.
method Existence results and a non-uniqueness example.
result Disproved a conjecture of Hamilton's about uniqueness.
We establish a new fundamental relationship between total curvature of knots and crossing number. If K is a smooth knot in 3-space, R the cross-section radius of a uniform tube neighborhood of K, L the arclength of K, and k the total curvature of K, then (up to a coefficient independent of K), crossing number of K < (k…
Study Transformer layers under cross-entropy training using mean field control.
problem Understanding the behavior of Transformer layers in cross-entropy training.
method Continuous-depth mean field control analysis, treating depth as time and layer parameters as controls.
result Derivation of a Pontryagin condition for the limiting population problem, involving the softmax residual.
The key condition A3w of Ma, Trudinger and Wang for regularity of optimal transportation maps is implied by the nonnegativity of a pseudo-Riemannian curvature -- which we call cross-curvature -- induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature nonnegativ…
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
Entropy measures geodesic flow complexity.
problem Measuring complexity of geodesic flows on manifolds.
method Introduced barcode entropy to measure exponential growth rate of not-too-short bars in Morse-theoretic barcodes.
result Barcode entropy bounds topological entropy and vice versa.
Graph cross network improves graph classification accuracy.
problem Improving graph classification accuracy.
method Graph cross network (GXN) with vertex infomax pooling (VIPool) and feature-crossing layer.
result Improves graph classification accuracy by 2.12% and 1.15%.
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
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.
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.
Paper proves uniqueness theorems for non-compact mean curvature flow.
problem Proving uniqueness for non-compact mean curvature flow.
method Energy argument and similar method for Ricci flows.
result Generalizes results by Chen and Yin on mean curvature flow with unbounded curvatures.