arXiv subjects

Joa Weber

Publications and source records attributed to Joa Weber.

32 records · Page 2Linked to original sources

arXiv2016

Classical Morse theory revisited I -- Backward $λ$-Lemma and homotopy type

We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in [2,3], as a dynamical thickening of the stable manifold. As a first application and to illustrate efficiency of the concept we reprove a fundamental theorem of classical Morse theory, Milnor's homotopical cell attachment theorem [1]. Dynamical thickening leads to a conceptually simple and short proof.

math.DS↗

arXiv2015

Contraction method and Lambda-Lemma

We reprove the $λ$-Lemma for finite dimensional gradient flows by generalizing the well-known contraction method proof of the local (un)stable manifold theorem. This only relies on the forward Cauchy problem. We obtain a rather quantitative description of (un)stable foliations which allows to equip each leaf with a copy of the flow on the central leaf -- the local (un)stable manifold. These dynamical thickenings are key tools in our recent work [Web]. The present paper provides their construction.

math.DS↗

arXiv2014

Stable foliations and semi-flow Morse homology

In case of the heat flow on the free loop space of a closed Riemannian manifold non-triviality of Morse homology for semi-flows is established by constructing a natural isomorphism to singular homology of the loop space. The construction is also new in finite dimensions. The main idea is to build a Morse filtration using Conley pairs and their pre-images under the time-$T$-map of the heat flow. A crucial step is to contract each Conley pair onto its part in the unstable manifold. To achieve this we construct stable foliations for Conley pairs using the recently found backward $λ$-Lemma [31]. These foliations are of independent interest [23].

math.DG↗

arXiv2014

A backward $λ$-Lemma for the forward heat flow

The inclination or $λ$-Lemma is a fundamental tool in finite dimensional hyperbolic dynamics. In contrast to finite dimension, we consider the forward semi-flow on the loop space of a closed Riemannian manifold $M$ provided by the heat flow. The main result is a backward $λ$-Lemma for the heat flow near a hyperbolic fixed point $x$. There are the following novelties. Firstly, infinite versus finite dimension. Secondly, semi-flow versus flow. Thirdly, suitable adaption provides a new proof in the finite dimensional case. Fourthly and a priori most surprisingly, our $λ$-Lemma moves the given disk transversal to the unstable manifold backward in time, although there is no backward flow. As a first application we propose a new method to calculate the Conley homotopy index of $x$.

math.AP↗

arXiv2013

An almost existence theorem for non-contractible periodic orbits in cotangent bundles

Assume M is a closed connected smooth manifold and H:T^*M->R a smooth proper function bounded from below. Suppose the sublevel set {H =d the level set {H=s} carries a periodic orbit z of the Hamiltonian system (T^*M,ω_0,H) representing α. Examples show that the condition that {H<d} contains M is necessary and almost existence cannot be improved to everywhere existence.

math.SG↗

arXiv2013

The backward λ-Lemma and Morse filtrations

Consider the infinite dimensional hyperbolic dynamical system provided by the (forward) heat semi-flow on the loop space of a closed Riemannian manifold M. We use the recently discovered backward λ-Lemma and elements of Conley theory to construct a Morse filtration of the loop space whose cellular filtration complex represents the Morse complex associated to the downward L2-gradient of the classical action functional. This paper is a survey. Details and proofs will be given in [6].

math.DS↗

arXiv2010

Morse homology for the heat flow

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 to nondegenerate closed geodesics of Morse index difference one.

math.DG↗

arXiv2009

A product estimate, the parabolic Weyl lemma and applications

We prove a product estimate that allows to estimate the quadratic first order nonlinearity of the harmonic map flow in the $L^p$ norm. Then the parabolic analogue of Weyl's lemma for the Lapace operator is established. Both results are applied to prove regularity for the heat flow by parabolic bootstrapping.

math.AP↗

arXiv2006

Three approaches towards Floer homology of cotangent bundles

Consider the cotangent bundle of a closed Riemannian manifold and an almost complex structure close to the one induced by the Riemannian metric. For Hamiltonians which grow for instance quadratically in the fibers outside of a compact set, one can define Floer homology and show that it is naturally isomorphic to singular homology of the free loop space. We review the three isomorphisms constructed by Viterbo (1996), Salamon-Weber (2003) and Abbondandolo-Schwarz (2004). The theory is illustrated by calculating Morse and Floer homology in case of the euclidean n-torus. Applications include existence of noncontractible periodic orbits of compactly supported Hamiltonians on open unit disc cotangent bundles which are sufficiently large over the zero section.

math.SG↗

arXiv2005

The Morse-Witten complex via dynamical systems

Given a smooth closed manifold M, the Morse-Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in [We-93] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman-Hartman theorem and the Lambda-Lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.

math.GT↗

arXiv2004

Noncontractible periodic orbits in cotangent bundles and Floer homology

For every nontrivial free homotopy class $α$ of loops in every closed connected Riemannian manifold $M$, we prove existence of a noncontractible 1-periodic orbit, for every compactly supported time-dependent Hamiltonian on the open unit cotangent bundle which is sufficiently large over the zero section. The proof shows that the Biran-Polterovich-Salamon capacity is finite for every closed connected Riemannian manifold and every free homotopy class of loops. This implies a dense existence theorem for periodic orbits on level hypersurfaces and, consequently, a refined version of the Weinstein conjecture: Existence of closed characteristics (one associated to each nontrivial $α$) on hypersurfaces in $T^*M$ which are of contact type and contain the zero section.

math.SG↗

arXiv2004

Floer homology and the heat flow

We study the heat flow in the loop space of a closed Riemannian manifold $M$ as an adiabatic limit of the Floer equations in the cotangent bundle. Our main application is a proof that the Floer homology of the cotangent bundle, for the Hamiltonian function kinetic plus potential energy, is naturally isomorphic to the homology of the loop space.

math.SG↗

arXiv2001

Morse theory on the loop space of flat tori and symplectic Floer theory

We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped with the natural symplectic structure. Both objects are isomorpic to the singular homology of the loop space. In an appendix we perturb the equations in order to eliminate degeneracies and to get to a situation with nondegenerate critical points only. The (co)homology groups turn out to be invariant under the perturbation.

dg-ga↗

arXiv2001

Perturbed closed geodesics are periodic orbits: Index and Transversality

We study the classical action functional $\SMC_V$ on the free loop space of a closed, finite dimensional Riemannian manifold $M$ and the symplectic action $\AMC_V$ on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in $M$. The potential $V\in C^\infty(M\times S^1,\R)$ serves as perturbation and we show that both functionals are Morse for generic $V$. In this case we prove that the Morse index of a critical point $x$ of $\SMC_V$ equals minus its Conley-Zehnder index when viewed as a critical point of $\AMC_V$ and if $x^*TM \to S^1$ is trivial. Otherwise a correction term +1 appears.

math.SG↗