 
Summary: Generalized Hopf differentials
Uwe Abresch
(joint work with Harold Rosenberg, Paris 7)
The basic global results in the theory of constant mean curvature (cmc) surfaces in
space forms are the theorems of A. D. Alexandrov and H. Hopf from the 1950ies [4,
8]. Alexandrov's theorem states that a closed, embedded cmc surface in S3
+, R3
,
or H3
space form is necessarily a standard distance sphere. Its proof is based
on a moving planes argument that is amazingly flexible and has been applied in
many other contexts since. It has even turned out to be fruitful for the theory of
nonlinear elliptic equations [7].
Hopf's theorem on the other hand states that an immersed cmc sphere in a
space form M3
is necessarily a standard distance sphere. The basic idea in Hopf's
argument is to observe that the (2, 0)part of the second fundamental form h =
. , A . of such a cmc surface 2
is a holomorphic quadratic differential, a fact
that is also one of the foundations of the theory of cmc tori in space forms [1, 5, 6].
