zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Constant mean curvature surface, harmonic maps, and universal Teichmüller space. (English) Zbl 0808.53056
T. K. Milnor [Trans. Am. Math. Soc. 280, 161-185 (1983; Zbl 0532.53047)]has proven that the Gauss map of a space-like surface in Minkowski three-space M 2,1 is harmonic if and only if the mean curvature of the surface is constant. Using this fact, the author constructs, for each holomorphic quadratic differential on a simply- connected domain Ω in , an injective harmonic map from Ω into the Poincaré disc D. This harmonic map is unique up to equivalent classes, provided it satisfies a completeness condition. Also, a classification of all complete hyperbolic space-like surfaces of constant mean curvature in M 2,1 has been found. M. Wolf [J. Differ. Geom. 29, No. 2, 449-479 (1989; Zbl 0655.58009)]shows that harmonic maps between surfaces are deeply related to the Teichmüller theory of compact Riemann surfaces. The author studies the analogy for the universal Teichmüller space and proves that a harmonic diffeomorphism from D onto itself is quasi-conformal if and only if the associated Hopf differential is bounded with respect to the Poincaré metric. As a consequence, a continuous map from the space of equivalence classes of holomorphic quadratic differentials under the action of the Möbius group, which is bounded with respect to the Poincaré metric, into the universal Teichmüller space has been obtained.

53C42Immersions (differential geometry)
58E20Harmonic maps between infinite-dimensional spaces