Joana Oliveira dos Santos

arXiv2017

Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry-Mather theory

We construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants developed by Oh and results by Abouzaid about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalization of graph selector. We then, following Bernard-Oliveira dos Santos, use these results to give a new characterization of the Aubry and Mane sets of a Tonelli Hamiltonian and to generalize a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.

math.SG↗

arXiv2015

Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in C^m

We prove two main results: (a) Suppose $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse special Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit 'Lawlor neck' family of examples found by Lawlor (Invent. math. 95, 1989). (b) Suppose $L$ is a closed, embedded, exact Lagrangian mean curvature flow expander in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit family of examples found by Joyce, Lee and Tsui, arXiv:0801.3721. If instead $L$ is immersed rather than embedded, the only extra possibility in (a),(b) is $L=Π_1\cupΠ_2$. Our methods, which are new and can probably be used to prove other similar uniqueness theorems, involve $J$-holomorphic curves, Lagrangian Floer cohomology, and Fukaya categories from symplectic topology. When $m=2$, (a) is easy to prove using hyperkahler geometry, and (b) is proved by Lotay and Neves, arXiv:1208.2729.

math.SG↗