Monotone Lagrangians in cotangent bundles of spheres

Abstract

We study the compact monotone Fukaya category of $T^{\ast }{S}^n$, for $n\ge 2$, and show that it is split-generated by two classes of objects: the zero-section $S^n$ (equipped with suitable bounding cochains) and a 1-parameter family of monotone Lagrangian tori $(S^1\times S^{n-1}{)}_{\tau }$, with monotonicity constants $\tau >0$ (equipped with rank 1 unitary local systems). As a consequence, any closed orientable spin monotone Lagrangian (possibly equipped with auxiliary data) with non-trivial Floer cohomology is non-displaceable from either $S^n$ or one of the $(S^1\times S^{n-1}{)}_{\tau }$. In the case of $T^{\ast }{S}^3$, the monotone Lagrangians $(S^1\times S^2{)}_{\tau }$ can be replaced by a family of monotone tori $T_{\tau }^3$.

37 pages, 7 figures. Several minor corrections and clarifications. Accepted in Advances in Mathematics