English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker{\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e.those (branched) maps effecting the conformal representation upon the disc of a{\it compact bordered Riemann surface}. The theory in question has somewell-known intersection with real algebraic geometry, especially Klein'sortho-symmetric curves via the paradigm of {\it total reality}. This leads to agallery of pictures quite pleasant to visit of which we have attempted to tracethe simplest representatives. This drifted us toward some electrodynamicmotions along real circuits of dividing curves perhaps reminiscent of Kepler'splanetary motions along ellipses. The ultimate origin of circle maps is ofcourse to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass.Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be foundin Klein (what we failed to assess on printed evidence), the pivotalcontribution belongs to Ahlfors 1950 supplying an existence-proof of circlemaps, as well as an analysis of an allied function-theoretic extremal problem.Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degreecontrols than available in Ahlfors' era. Accordingly, our partisan belief isthat much remains to be clarified regarding the foundation and optimal controlof Ahlfors circle maps. The game of sharp estimation may look narrow-minded"Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual tocontemplate how conformal and algebraic geometry are fighting together for thesoul of Riemann surfaces. A second part explores the connection with Hilbert's16th as envisioned by Rohlin 1978.

Ahlfors circle maps and total reality: from Riemann to Rohlin