Extremal analytic discs with prescribed boundary data

Abstract:

This paper concerns the existence and uniqueness of extremal analytic discs with prescribed boundary data in a bounded strictly linearly convex domain $D$ in ${{\mathbf {C}}^n}$. We prove that for any two distinct points $p$, $q$ in $\partial D$ (respectively, $p \in \partial D$ and a vector $v$ such that $\sqrt { - 1} v \in {T_p}(\partial D)$ and $\langle v, \overline \nu (p)\rangle = \sum \nolimits _1^n {{v_j}{{\overline \nu }_j}(p) > 0}$ where $\nu (p)$ is the outward normal to $\partial D$ at $p$) there exists an extremal analytic disc $f$ passing through $p$, $q$ if $\partial D \in {C^k}$, $k \geqslant 3$ (respectively, $f(1) = p$, $f’ (1) = v$ if $\partial D \in {C^k}$, $k \geqslant 14$). Consequently, we can foliate $\overline D$ with these extremal analytic discs.