Let $L\subset T^\ast S^n$ be a properly embedded Lagrangian submanifold homeomorphic to $\mathbb{R}^n$ with respect to the canonical symplectic structure on $T^\ast S^n$, and suppose $L$ intersects the zero section $S^n$ at exactly one point. Is it necessarily true that $L$ is Hamiltonian isotopic to a cotangent fiber?

## 1 Answer

Let $\tau \in \mathrm{Symp}^c(T^*S^n)$ denote the generalised Dehn twist. Seidel has proven using Floer homology that $\tau^k(T_x^*S^n)$, $k \neq 0$, never is compactly supported Hamiltonian isotopic to the fibre $T_x^*S^n$. In some cases (e.g. $n=2$) there are no classical obstructions to this.

You could ask the question under the additional assumption that the Lagrangian is contained in an arbitrarily small neighbourhood of the fibre $T^*_xS^n$; in this situation the answer to the problem is (to my understanding) known only in the cases $n=1,2$, where the case $n=2$ is due to Eliashberg-Polterovich.

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