arXiv subjects

Johannes Hauber

Publications and source records attributed to Johannes Hauber.

4 recordsLinked to original sources

arXiv2026

The nearby Lagrangian conjecture for pinwheels

The Lagrangian skeleton of the rational homology ball $B_{p,q}$, for $0<q<p$ coprime integers, is an immersed but not embedded Lagrangian, called a $(p,q)$-pinwheel. We show that any two embeddings of Lagrangian $(p,q)$-pinwheels in $B_{p,q}$ are related by a compactly supported Hamiltonian isotopy, establishing Arnold's nearby Lagrangian conjecture for this wide class of singular Lagrangians. Our proof has two largely independent parts: the first uses neck-stretching and the symplectic rational blow-up to understand embeddings of pinwheels up to symplectomorphism; the second computes that $\text{Symp}_c(B_{p,q})$ is generated by a twist about the pinwheel, which we call the pintwist $τ_{p,q}$. We provide three applications of our methods: Gromov non-squeezing for pin-balls; a new proof of the local Lagrangian unknotting theorem of Eliashberg--Polterovich; and that the only Lagrangian $(n,m)$-pinwheel in $B_{p,q}$ is of type $(p,q)$.

math.SG↗

arXiv2025

Markov staircases

Rational homology ellipsoids are certain Liouville domains diffeomorphic to rational homology balls and having Lagrangian pin-wheels as their skeleta. From the point of view of almost toric fibrations, they are a natural generalisation of usual symplectic ellipsoids. We study symplectic embeddings of rational homology ellipsoids into the complex projective plane and we show that for each Markov triple, this problem gives rise to an infinite staircase. A key ingredient in the proof is the result that any two such embeddings are Hamiltonian isotopic. We also prove constraints on sizes for pairs of disjoint embeddings.

math.SG↗

arXiv2025

Pinwheels in symplectic rational and ruled surfaces and non-squeezing of rational homology balls

We use almost toric fibrations and the symplectic rational blow-up to determine when certain Lagrangian pinwheels, which we call liminal, embed in symplectic rational and ruled surfaces. The case of $L_{2,1}$-pinwheels, namely Lagrangian $\mathbb{R}P^2$'s, answers a question of Kronheimer in the negative, exhibiting a symplectic non-spin $4$-manifold that does not carry a Lagrangian $\mathbb{R}P^2$. In addition, we provide applications to symplectic embeddings of rational homology balls. In particular, we generalize Gromov's classical non-squeezing theorem by proving that a rational homology ball $B_{n,1}(1)$ embeds into the rational homology cylinder $B_{n,1}(α,\infty)$ if and only if $α\geq 1$. Along the way, we prove various properties of Lagrangian pinwheels of independent interest, such as describing their homological complement, providing a short proof that performing a symplectic rational blow-up of a Lagrangian pinwheel in a positive symplectic rational manifold yields a symplectic manifold which is also rational, and showing a self-intersection formula for Lagrangian pinwheels.

math.SG↗

arXiv2024

Semi-Local Exotic Lagrangian Tori in Dimension Four

We study exotic Lagrangian tori in dimension four. In certain Stein domains $B_{dpq}$ (which naturally appear in almost toric fibrations) we find $d+1$ families of monotone Lagrangian tori which are mutually distinct, up to symplectomorphisms. We prove that these remain distinct under embeddings of $B_{dpq}$ into geometrically bounded symplectic four-manifolds. We show that there are infinitely many different such embeddings when $X$ is compact and (almost) toric and hence conclude that $X$ contains arbitrarily many Lagrangian tori which are distinct up to symplectomorphisms of $X$. In dimension four arbitrarily many different Lagrangian tori were previously known only in del Pezzo surfaces. Neither the embedded tori, nor the ambient space $X$ needs to be monotone for our methods to work.

math.SG↗