From Teichmüller to Shoen–Yau: Extremal mappings between Riemann surfaces
Abstract: There are two now classical descriptions of the moduli space of a Riemann surface via the theory of extremal mappings. The first from Teichmu ̈ller in the 1940s (rigorously es- tablished by Ahlfors in 1953) and through the existence of extremal quasiconformal mappings. The second is through Schoen-Yau’s existence theory for unique harmonic diffeomorphisms in the 1970s, and developed into a theory of moduli by many, including Wolf, Tromba and Wolpert many years later. The important ingredient in both is the existence of a holomorphic quadratic differential, from the Beltrami coefficient of an extremal quasiconformal mapping (Teichmu ̈ller) or from the Hopf equation (Harmonic). These quadratic differentials define the cotangent space to the moduli space. Here we show that in fact both of these approaches are manifestations of the same theory (that of existence of diffeomorphic extremal mappings of finite distortion) in limiting regimes. We identify parameterised families of moduli spaces (Beltrami coe…
Event Type: Seminar. Research Area: Analysis and Differential Geometry. Location: MSC E408. Speaker Name: Professor Gaven Martin. Speaker Institution: Massey University (New Zealand).
Wednesday, October 22, 2025, 4:00 PM – 5:00 PM.