Abstract
Gustafsson and Lin recently published a significant result concerning Laplacian growth problems that start from a simply connected planar domain. However, the validity of their result depends on the verification of a particular conjecture. This paper provides the missing proof.
Highlights
A recent book of Gustafsson and Lin [4] explores the evolution of domains under a Laplacian growth process that starts from a connected planar domain with smooth boundary
A key result of theirs, Theorem 5.1, states that this process can be continued indefinitely as a family of connected domains on a suitable branched Riemann surface. Their theorem relies on the validity of a lemma which they believe to be true but are unable to prove. (See section 8 of [3].) The purpose of this note is to verify their conjecture and so complete the proof of their result
Let g be a holomorphic function on a connected neighbourhood ω of D, where D denotes the unit disc, and let λ denote planar Lebesgue measure
Summary
A recent book of Gustafsson and Lin [4] explores the evolution of domains under a Laplacian growth process that starts from a connected planar domain with smooth boundary. A key result of theirs, Theorem 5.1, states that this process can be continued indefinitely as a family of connected domains on a suitable branched Riemann surface. Their theorem relies on the validity of a lemma which they believe to be true but are unable to prove. Let g be a holomorphic function on a connected neighbourhood ω of D, where D denotes the unit disc, and let λ denote planar Lebesgue measure.
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