2022/11/07 by Matsuda, Ryo
#30F60 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2211.03290
Teichmüller space Teich(R) of a Riemann surface R is a deformation space of R. In this paper, we prove a sufficient condition for extremality of the Beltrami coefficients when R has the \mathbb Z action. As an application, we discuss the construction of geodesics. Earle-Kra-Krushkaĺ proved that the necessary and sufficient conditions for the geodesics connecting [0] and [μ] to be unique are ‖ μ0 ‖∞ = | μ0 | ( z ) (a.e.z) and ``unique extremality''. As a byproduct of our results, we show that we cannot exclude ``unique extremality''.To show the above claim, we construct a point [μ0] in Teich(\mathbb C ∖ \mathbb Z), satisfying ‖ μ0 ‖∞ = | μ0 | ( z ) (a.e.z) and there exists a family of geodesics \ γλ\ λ∈ D connecting [0] and [μ0] with complex analytic parameter, where D is an open set in l∞.