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)
Harmonic diffeomorphisms between complete Riemann surfaces of negative curvature. (English) Zbl 1200.58010

The Teichmüller space for an open manifold was introduced by the second author in [Asian J. Math. 2, No. 2, 355–404 (1998; Zbl 1045.58006)] and was studied in detail for Riemann surfaces.

In the paper under review, the authors investigate harmonic diffeomorphisms between complete Riemann surfaces of negative curvature. More precisely, besides many other results, they prove the following:

Let (M 2 ,g 0 ) and (M 2 ,g 1 ) be two open Riemann surfaces. Let {g t } 0t1 be a curve in the Sobolov topology, K g t -1, infσ(Δ 0 (g t ))>0, r inj (g t )>0, 0t1. Then there exists a unique harmonic diffeomorphism f 1 :(M 2 ,g 0 )(M 2 ,g 1 ) which is isotopic by harmonic diffeomorphisms to id:(M 2 ,g 0 )(M 2 ,g 0 ) in the unit component 𝒟 0 r+1 of the completed diffeomorphism group 𝒟 r+1 .

The technique essentially relies on the framework of non-linear Sobolov analysis on open manifolds as developed by the second author in [Global analysis on open manifolds. New York, NY: Nova Science Publishers (2007; Zbl 1188.58001)]. Application to Teichmüller theory for open surfaces is also given.

58D27Moduli problems for differential geometric structures on spaces of mappings
58E20Harmonic maps between infinite-dimensional spaces
30F60Teichmüller theory