Title: Real-analytic submanifolds which are local uniqueness sets for holomorphic functions of 𝐶³
Abstract: The following problem is considered. Given a real-analytic two-dimensional submanifold, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of complex Euclidean three-space, are ambient holomorphic functions determined by their values on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M question-mark"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>?</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M?</mml:annotation> </mml:semantics> </mml:math> </inline-formula> For a large class of submanifolds a necessary and sufficient condition is found for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be a local uniqueness set for holomorphic functions on complex three-space. Finally, the general problem is shown to be related to two-dimensional Nevanlinna theory.