Morse Index of Free Boundary Disk in Pseudoconvex Domain

Abstract

AbstractIn this paper, we study the Morse index for the $$\overline{\partial }$$ ∂ ¯ -energy of a non-holomorphic disk in a strictly pseudoconvex domain in $$\mathbb {C}^n$$ 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 $$n-1$$ n - 1 . We also extend the result to domains which satisfy the weaker k-pseudoconvexity property for $$k\ge 2$$ k ≥ 2 .