arXiv research

Proves stability of geodesic flows on closed surfaces.

The article calculates the $\mathbb{F}$-convergence rate for Ricci flows with closed and smooth tangent flows.

The $(α,β)$-Ricci-Yamabe flow exists on closed manifolds.

The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.

The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.

The paper proves the existence of surfaces of section for geodesic flows on closed surfaces.

New results on geodesic flows using curve shortening flow.

Homoclinic orbits found in geodesic flows on surfaces.

The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.

We produce infinitely many examples of Anosov flows in closed 3-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one …

New examples of mean curvature flow converge to minimal surfaces with multiplicity 2.

We prove Calegari's conjecture that every quasigeodesic flow on a closed hyperbolic 3-manifold has closed orbits.

Ricci flow modelled on specific singularities on closed manifolds.

The paper proves smoothness of mean curvature flow for generic initial data in 3D and 4D.

Higher KdV flows on spaces of closed equicentroaffine plane curves are studied and it is shown that the flows are described as certain multi-Hamiltonian systems on the spaces. Multi-Hamiltonian systems describing higher mKdV flows are also given on spaces of closed Euclidean plane curves via the geometric Miura transfo…

We consider the closed orbit structure of generic gradient flows of Morse closed 1-forms. The torsion of a chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex of the universal cover detects the eta function of the flow. We extend this result to arbitrary Morse closed 1-form…

The paper studies Ricci flow with finite curvature integrals on manifolds.

Two flows for convex curves converge to circles smoothly.

Entropy rigidity for Finsler flows but collapse for Reeb flows.

We characterize the conjugate linearized Ricci flow and the associated backward heat kernel on closed three--manifolds of bounded geometry. We discuss their properties, and introduce the notion of Ricci flow conjugated constraint sets which characterizes a way of Ricci flow averaging metric dependent geometrical data. …

Study shows generic surfaces avoid complex flow patterns.

We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that point. That is the limit of the parabolic rescalings does not depend on the chose…

Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.

The main result in this paper is the $C^{\infty}$ closing lemma for a large family of Hamiltonian flows on $4$-dimensional symplectic manifolds, which includes classical Hamiltonian systems. First we prove the $C^{\infty}$ closing lemma and the $C^r$ general density theorem for geodesic flows on closed Finsler surfaces…

Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.

In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the C^infty-topology to a simple circle. Our results yield a characterisation of …

The paper classifies flows of ancient curves in 2D space.

In this paper, we study two kind of L^2 norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability and asymptotic behavior to such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.

Kähler-Ricci flow smooths out positive closed currents with divisorial singularities

Paper proves unique contact structure supported by positive flow-spines.

We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically…

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $α$-Gauss c…

Sharp convergence rate for curvature stability in planar free elastic flow.

The study finds infinitely many periodic orbits that can be used to modify Anosov flows.

Proves mean curvature flow from conical singularities to shrinkers.

Closed-form flow matching yields similar performance to stochastic version, improving model performance.

We consider the flows generated by generic gradients of Morse maps of a closed connected manifold $M$ to a circle. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called…

The paper improves the regularity and existence of pseudo Calabi flow.

In this paper we prove that the generic singularities of mean curvature flow of closed embedded surfaces in $\mathbb R^3$ modeled by closed self-shrinkers with multiplicity has multiplicity one. Together with the previous result by Colding-Minicozzi in [CM12], we conclude that the only generic singularity of mean curva…

For any $α>0,$ we study $k^α$-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $C^{\infty}$-norm.$\$Other relevant $k^α$-type nonlocal flow is also discussed when $α\geq1.\$

The study examines the long-term behavior of mean curvature flows in closed 3-manifolds.

The paper introduces branched α-flows on surfaces with negative Euler characteristic and proves their long-term existence and convergence.

Anosov flows in hyperbolic 3-manifolds are quasigeodesic if not R-covered.

We study a normalized version of the second order renormalization group flow on closed Riemannian surfaces. We discuss some general properties of this flow and establish several basic formulas. In particular, we focus on surfaces with zero and positive Euler characteristic.

Establishes smooth Ricci flows from convex surfaces in 3D space.

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.

We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces

Researchers create nonconvex, non-soliton ancient flows in various dimensions.