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Critical Points of Green's Function and Geometric Function Theory
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-3125-3030
2012 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 61, no 3, 939-1017 p.Article in journal (Refereed) Published
Abstract [en]

We study questions related to critical points of the Green's function of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's function are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding Mobius invariant differential operators. In the doubly connected case the three Eisenstein series E-2, E-4, E-6 are used. A specific result is that a doubly connected domain is the disjoint union of the set of critical points of the Green's function, the set of zeros of the Bergman kernel and the separating boundary limit positions for these.

Place, publisher, year, edition, pages
2012. Vol. 61, no 3, 939-1017 p.
Keyword [en]
critical point, Green's function, Neumann function, Bergman kernel, Schiffer kernel, Schottky-Klein prime form, Schottky double, weighted Bergman space, Poincare metric, Martin boundary, projective structure, projective connection, affine connection, Eisenstein series
National Category
URN: urn:nbn:se:kth:diva-125796DOI: 10.1512/iumj.2012.61.4621ISI: 000321231000003ScopusID: 2-s2.0-84880872132OAI: diva2:640851
Swedish Research Council

QC 20130814

Available from: 2013-08-14 Created: 2013-08-13 Last updated: 2013-08-14Bibliographically approved

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Gustafsson, Björn
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