Morse Index of Free Boundary Disk in Pseudoconvex Domain

Abstract

In this paper, we study the Morse index for the overline{partial }-energy of a non-holomorphic disk in a strictly pseudoconvex domain in mathbb {C}^n or in a Kähler manifold with non-negative bisectional curvature. We give a proof that a overline{partial }-energy minimizing disk is holomorphic; in fact, more generally we show that a non-holomorphic critical disk for the overline{partial }-energy has Morse index at least n-1. We also extend the result to domains which satisfy the weaker k-pseudoconvexity property for kge 2.