Morse index of free boundary disk for pseudoconvex domain

In this paper we study the Morse index for the $\bar{\mathrm{\partial }}$-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 $\bar{\mathrm{\partial }}$-energy minimizing disk is holomorphic; in fact, more generally we show that a non-holomorphic critical disk for the $\bar{\mathrm{\partial }}$-energy has Morse index at least $n-1$. We also extend the result to domains which satisfy the weaker $k$-pseudoconvexity property for $k\ge 2$.