arXiv research

New method to classify simple Smale flows on $S^3$.

In this paper, following J. Franks' work on Lyapunov graphs of nonsingular Smale flows on $S^3$, we study Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$. More precisely, we determine necessary and sufficient conditions on an abstract Lyapunov graph to be associated with a nonsingular Smale flow on $S^1 …

A Smale flow is a structurally stable flow with one dimensional invariant sets. We use information from homology and template theory to construct, visualize and in some cases, classify, nonsingular Smale flows in the 3-sphere.

Defines and calculates foliation homology from flows.

Homoclinic orbits found in geodesic flows on surfaces.

We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic fixed point in a flow on $\real^3$ originally described by Shil'nikov. By modifying how the invariant manifolds wrap around themselves, or ``pleat,'' we may apply the theory of templates, or branched …

The paper proves a conjecture linking two metrics on manifold cohomology.

New proof of Smale conjecture for RP^3 and lens spaces using min-max theory.

Combines techniques to remove tameness condition in Morse-Smale flows.

The article recovers the Smale conjecture on a Sasakian 3-sphere using Legendrian mean curvature flow.

Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.

For a closed oriented 3-manifold $Y$ we define $n(Y)$ to be the minimal non-negative number such that in each homotopy class of non-singular vector fields of $Y$ there is a Morse-Smale vector field with less or equal to $n(Y)$ periodic orbits. We combine the construction process of Morse-Smale flows given in [2] with h…

The paper studies 3-manifolds with specific Morse-Smale diffeomorphisms and finds they are homeomorphic to lens spaces.

We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the ho…

In this paper, we first develope the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tool…

A Morse complex for Axiom A flows on smooth manifolds.

Horizon saddle connections imply dense hyperbolic geodesics on dilation surfaces.

We consider the dynamics of vector fields on three-manifolds which are constrained to lie within a plane field, such as occurs in nonholonomic dynamics. On compact manifolds, such vector fields force dynamics beyond that of a gradient flow, except in cases where the underlying manifold is topologically simple. Furtherm…

The abstract proves that certain Reeb vector fields on 3-manifolds have Birkhoff sections.

In this paper, we first discuss the regular level set of a nonsingular Smale flow (NSF) on a 3-manifold. The main result about this topic is that a 3-manifold $M$ admits an NSF flow which has a regular level set homeomorphic to $(n+1)T^{2}$ $(n\in \mathbb{Z}, n\geq 0)$ if and only if $M=M'\sharp n S^{1}\times S^{2}$. T…

We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point o…

The Palais-Smale condition is proven for various knot energies.

On a smooth, compact and oriented manifold without boundary, we give a complete description of the correlation function of a Morse-Smale gradient flow satisfying a certain nonresonance assumption. This is done by analyzing precisely the spectrum of the generator of such a flow acting on certain anisotropic spaces of cu…

Determines regular homotopy classes for link immersions of simple singularities.

We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic pi…

We consider a Morse function $f$ and a Morse-Smale gradient-like vector field $X$ on a compact connected oriented 3-manifold $M$ such that $f$ has only one critical point of index 3. Based on Laudenbach's ideas, we will show that the flow of $X$ can be isotoped into one so that the trajectory spaces of the new flow pro…

The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $Φ: \mathbb{R}\times X\to X$ on pfaffian set $X$ is tame if the graph of $Φ$ is a pfaffian subset of $\mathbb{R}\times X\times X$. Any compact tame set admits plenty tame flows. We prove …

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for i…

We prove a lower bound on the number of maximally broken trajectories of the negative gradient flow of a Morse-Smale function on a closed aspherical manifold in terms of integral (torsion) homology.

The paper proves critical point results for Frechet manifolds.

We introduce an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra, and on its Hessenberg-type submanifolds. In our local coordinates the Toda flow becomes linear. We use these new coordinates to show that the Toda flow on the manifold of full flags is Morse-Smal…

In the first part of this paper, we consider smooth maps from a compact orientable 3-manifold without boundary to the 2-sphere. We give a geometric criterion to decide whether two given maps are homotopic, based on the sets of points where the maps are equal or antipodal. We extend this criterion to non-singular vector…

We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies…

In 1969 M. Gromov in his PhD thesis greatly generalized Smale-Hirsch-Phillips immersion-submersion theory by proving what is now called the h-principle for invariant open differential relations over open manifolds. Gromov extracted the original geometric idea of Smale and put it to work in the maximal possible generali…

Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.

We give in this paper bounds for the Morse indices of a large class of simple geodesics on a surface with a generic metric. To our knowledge these bounds are the first that use only the generic hypothesis on the metric.

We consider general Morse-Smale diffeomorphisms on a closed orientable two-dimentional surface. In this paper it is proved that the complete topological invariant of Morse-Smale diffeomorphisms is finite, the algorithm of the construction of the complete topological invariant in explicit form is given and necessary and…

The Morse-Smale complex of a function $f$ decomposes the sample space into cells where $f$ is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize, and compare multivariate functions. In this paper, we present some statistical results on es…

We define the geometric complex associated to a Morse-Bott-Smale vector field, cf. [Austin-Braam, 1995], and its associated spectral sequence. We prove an extension of the Bismut-Zhang theorem to Morse-Bott-Smale functions. The proof is based on the Bismut-Zhang theorem for Morse-Smale functions, see [Bismut-Zhang, 199…

We generalize some of the results of Harvey, Lawson and Latschev about transgression formulas. The focus here is on flowing forms via vertical vector fields, especially Morse-Bott-Smale vector fields. We prove a very general transgression formula including also a version covering non-compact situations. Among applicati…

In this note, we construct new examples of Lorentzian Sasaki-Einstein (LSE) metrics on Smale manifolds $M.$ It has already been established in \cite{Gmz2} that such metrics exist on the so-called torsion free Smale manifolds, i.e. the $k$-fold connected sum of $S^{2}\times S^{3}.$ Now, we show that LSE metrics exist on…

Study Palais-Smale sequences for Ricci curvature on homogeneous spaces.

Paper constructs Thom-Smale complex using instantons from Morse functions.

We prove the compactness of solutions to general fourth order elliptic equations which are L^1-perturbations of the Q-curvature equation on compact Riemannian 4-maniods. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by product, we …

Disproves the Smale Conjecture for S^4 by showing Diff(S^4) is not SO(5).

Classifies Morse flows on 3-sphere with specific saddle connections.

Study on Morse homology for reflection actions on manifolds.

The original Smale Conjecture asserted that the inclusion of the group O(4) of isometries of the round 3-sphere S into the full diffeomorphism group Diff(S) is a homotopy equivalence. The (Generalized) Smale Conjecture asserts that the inclusion of Isom(M) into Diff(M) is a homotopy equivalence whenever M is an ellipti…