Quadratic Differentials, Quaternionic Forms, and Surfaces

In this paper we address three problems stemming from the fundamental question: Let M be an oriented surface, how much data do we need to identify an immersion f : M → R or its shape? A generic immersion f is determined up to homothety and translation by its conformal class and its tangent plane map Tf : M → Gr(2, 3) whose value Tf (p) at each point p ∈ M is the plane tangent to the image surface f(M). There are exceptions, called Christoffel immersions. Their classification is known as Christoffel’s problem. The problem was posed and to a large extent solved locally in [5]. A short complete solution of Christoffel’s problem, including closed surfaces, is presented in Section 4. Bonnet noticed in [3] that if two immersions induce the same first fundamental form and the same mean curvature function then they are congruent unless they are quite exceptional. Bonnet’s problem is to classify all such exceptional immersions. A local classification of the umbilic-free and the constant mean curvature exceptions follows from [3, 4, 6]. Until now it was not known whether there exist other exceptions. The existence and description of the new examples is discussed in Section 5. Christoffel’s and Bonnet’s problems inevitably lead to the study of isothermic immersions [7, 13]. An umbilic-free immersion is called umbilic-free classical isothermic immersion if every point p ∈ M admits isothermal local coordinates in which its second fundamental form is diagonal. There exist classical definitions of isothermic immersions with umbilic points but no definitive one. Up to now it was not clear how to motivate and define global isothermic immersions. In this paper we use a hitherto unsuspected connection between isothermic immersions and quadratic differentials to study global isothermic immersions and immersions with umbilic points, branch points, and ends (see Section 3). The presented theory is designed to solve the targeted geometric problems and to resolve some quirks in the folklore surrounding the classical isothermic immersions. The main tool used in this paper is the quaternionic calculus on surfaces [13]. The necessary material is summarized in Section 2. Some of the results in this paper were announced in [15]. Unless explicitly specified otherwise all immersions in this paper are assumed to be at least C. On the other hand many of the constructions and arguments do not need much smoothness. Weaker smoothness requirements, regularity, and partial regularity are discussed in [12].