Asymptotic behavior of exotic lagrangian tori ta,b,c in ℂp2 as a+b+c → ∞

Abstract

© 2021, International Press, Inc.. All rights reserved.In this paper, we study various asymptotic behavior of the infinite family of monotone Lagrangian tori Ta,b,c in ℂP2 associated to Markov triples (a,b,c) described in [Via16]. We first prove that the Gromov capacity of the complement ℂP2 \Ta,b,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 {Ta,b,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.