Asymptotic behavior of exotic Lagrangian tori T-a,T-b,T-c in CP2 as a plus b plus c -> infinity

Abstract

In this paper, we study various asymptotic behavior of the infinite family of monotone Lagrangian tori T-a,T-b,T-c in CP2 associated to Markov triples (a, b, c) described in [Via16]. We first prove that the Gromov capacity of the complement CP2\T-a,T-b,T-c is greater than or equal to 1/3 of the area of the complex line for all Markov triple (a, b, c). We then prove that there is a representative of the family {T-a,T-b,T-c} whose loci completely miss a metric ball of nonzero size and in particular the loci of the union of the family is not dense in CP2.