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arXiv · 2403.00408

Semi-Local Exotic Lagrangian Tori in Dimension Four

Joé Brendel·Johannes Hauber·Joel Schmitz
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Abstract

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.

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Joé Brendel, Johannes Hauber, Joel Schmitz. 2024-10-09. Semi-Local Exotic Lagrangian Tori in Dimension Four. https://arxiv.org/abs/2403.00408

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