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Kai Zehmisch

Publications and source records attributed to Kai Zehmisch.

At least 19 recordsLinked to original sources
arXiv2026

A contact-geometric variant of Smale's contraction

Motivated by Smale's contraction of the compactly supported diffeomorphism group of the two-disc, we study contact forms on an open rotationally symmetric Darboux ball that agree with the standard contact form outside a compact set and whose Reeb flows have no trapped orbits. Such forms are called vertically convex. In dimensions three and five, we identify the quotient of their space by compactly supported diffeomorphisms with a contractible monodromy space and prove an equivariant product splitting. Consequently, the orbit of the standard contact form is a strong deformation retract. It follows that the space of vertically convex contact forms is contractible in dimension three and homotopy equivalent to the compactly supported diffeomorphism group, hence connected, in dimension five. The analogous subspace of forms defining the standard contact structure has the homotopy type of the corresponding compactly supported contactomorphism group in dimension five and is contractible in dimension three.

math.SG↗
arXiv2026

Contactomorphic vertically convex domains

We consider the standard Darboux space equipped with the radial symmetric contact form. We study co-orientation preserving contactomorphisms between relatively compact domains up to the boundary. We determine the contactomorphism classes among all strict vertically convex domains over a round ball in the Liouville hyperplane that are radially symmetric about the Reeb axis and whose boundary coincide along a neighbourhood of the common equator. The total invariant is the mean curvature of the bounding sphere at the umbilic points with the same sign. Replacing the Liouville hyperplane by codisc bundles of closed non-Besse Riemannian manifolds or finite symplectisations of closed non-Besse strict contact manifolds analogous results are formulated in terms of characteristic length and total characteristic action, resp.

math.SG↗
arXiv2025

Fitting without fittings

We show that all symplectically aspherical fillings of the unit cotangent bundle of a given odd-dimensional sphere are diffeomorphic to the corresponding unit co-disc bundle. The concept of fittings previously introduced is not needed.

math.SG↗
arXiv2024

Non-fillability of overtwisted contact manifolds via polyfolds

We prove that any weakly symplectically fillable contact manifold is tight. Furthermore we verify the strong Weinstein conjecture for contact manifolds that appear as the concave boundary of a directed symplectic cobordism whose positive boundary satisfies the weak-filling condition and is overtwisted. Similar results are obtained in the presence of bordered Legendrian open books whose binding-complement has vanishing second Stiefel-Whitney class. The results are obtained via polyfolds.

math.SG↗
arXiv2024

Symplectic capacities of disc cotangent bundles of flat tori

We show that on the unit disc cotangent bundle of flat Riemannian tori, all normalized capacities coincide with twice the systole. The same result holds for flat, reversible Finsler tori and normalized capacities that are greater than or equal to the Hofer-Zehnder capacity.

math.SG↗
arXiv2022

Why bootstrapping for $J$-holomorphic curves fails in $C^k$

We present a simple example for the failure of the Calderón-Zygmund estimate for the $\bar{\partial}$-operator when the Sobolev $(k,p)$-norms are replaced by the $C^k$-norms. This example is discussed in the context of elliptic bootstrapping, Fredholm theory, and the regularity of $J$-holomorphic curves.

math.AP↗
arXiv2021

Subcritical polarisations of symplectic manifolds have degree one

We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by Biran and Cieliebak on subcritical polarisations of symplectic manifolds. Our proof is based on a simple homological argument using ideas of Kulkarni-Wood.

math.SG↗
arXiv2019

Diffeomorphism type via aperiodicity in Reeb dynamics

We characterise boundary shaped disc like neighbourhoods of certain isotropic submanifolds in terms of aperiodicity of Reeb flows. We prove uniqueness of homotopy and diffeomorphism type of such contact manifolds assuming non-existence of short periodic Reeb orbits.

math.SG↗
arXiv2019

A symplectic dynamics proof of the degree-genus formula

We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere, defining the Hopf fibration. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

math.DS↗
arXiv2019

Pseudorotations of the 2-disc and Reeb flows on the 3-sphere

We use Lerman's contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed 3-manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the 2-disc constructed by Fayad-Katok embed into the Reeb flow of a dynamically convex contact form on the 3-sphere.

math.DS↗
arXiv2019

Symplectic dynamics and the 3-sphere

Given a knot in a closed connected orientable 3-manifold we prove that if the exterior of the knot admits an aperiodic contact form that is Euclidean near the boundary, then the 3-manifold is diffeomorphic to the 3-sphere and the knot is the unknot.

math.SG↗
arXiv2018

Symplectic dynamics of contact isotropic torus complements

We determine the homotopy type of isotropic torus complements in closed contact manifolds in terms of Reeb dynamics of special contact forms. For that we utilize holomorphic curve techniques known from symplectic field theory as Gromov-Hofer compactness and localized transversality on non-compact contact manifolds.

math.SG↗
arXiv2018

Periodic orbits in virtually contact structures

We prove that certain non-exact magnetic Hamiltonian systems on products of closed hyperbolic surfaces and with a potential function of large oscillation admit non-constant contractible periodic solutions of energy below the Mañé critical value. For that we develop a theory of holomorphic curves in symplectizations of non-compact contact manifolds that arise as the covering space of a virtually contact structure whose contact form is bounded with all derivatives up to order three.

math.SG↗
arXiv2017

Odd-symplectic forms via surgery and minimality in symplectic dynamics

We construct an infinite family of odd-symplectic forms (also known as Hamiltonian structures) on the 3-sphere that do not admit a symplectic cobordism to the standard contact structure on the 3-sphere. This answers in the negative a question raised by Joel Fish motivated by the search for minimal characteristic flows.

math.DS↗