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 D in C n {{\mathbf {C}}^n} . We prove that for any two distinct points p p , q q in ∂ D \partial D (respectively, p ∈ ∂ D p \in \partial D and a vector v v such that − 1 v ∈ T p ( ∂ D ) \sqrt { - 1} v \in {T_p}(\partial D) and ⟨ v , ν ¯ ( p ) ⟩ = ∑ 1 n v j ν ¯ j ( p ) > 0 \langle v,\,\overline \nu (p)\rangle = \sum \nolimits _1^n {{v_j}{{\overline \nu }_j}(p) > 0} where ν ( p ) \nu (p) is the outward normal to ∂ D \partial D at p p ) there exists an extremal analytic disc f f passing through p p , q q if ∂ D ∈ C k \partial D \in {C^k} , k ⩾ 3 k \geqslant 3 (respectively, f ( 1 ) = p f(1) = p , f ′ ( 1 ) = v f’ (1) = v if ∂ D ∈ C k \partial D \in {C^k} , k ⩾ 14 k \geqslant 14 ). Consequently, we can foliate D ¯ \overline D with these extremal analytic discs.