zbMATH — the first resource for mathematics

Examples
 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.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 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)
Harmonic maps from the complex plane into surfaces with nonpositive curvature. (English) Zbl 0843.58028
Summary: We give a characterization for an orientation preserving harmonic diffeomorphism from $ℂ$ into a complete, simply connected, negatively pinched surface to have a polynomial growth Hopf differential. In particular, we prove that an orientation preserving harmonic diffeomorphism from $ℂ$ into the Poincaré disk $ℍ$ has a polynomial growth Hopf differential of degree $m$ if and only if its image is an ideal polygon with $m+2$ vertices on $\partial ℍ$, with the assumption that the conformal metric on $ℂ$ with the $\partial$-energy density as the conformal factor is complete. We will describe the geometric behavior of this harmonic diffeomorphisms in terms of the trajectories of their Hopf differentials. We will also construct all harmonic diffeomorphisms in this class, and prove that there is an $m-1$ parameter family of nontrivially distinct harmonic diffeomorphisms from the complex plane to a fixed ideal polygon with $m+2$ vertices in the hyperbolic plane. In particular, such harmonic maps are not unique, answering a question of Schoen.

MSC:
 58E20 Harmonic maps between infinite-dimensional spaces 53C42 Immersions (differential geometry) 53C50 Lorentz manifolds, manifolds with indefinite metrics