Geodesic
Construction of harmonic diffeomorphisms and minimal graphs
Pascal Collin1 ; Harold Rosenberg2
1 Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne<br/>F-31062 Toulouse Cedex 9<br/>France
2 IMPA<br/>Estrada Dona Castorina 110<br/>Rio de Janeiro<br/>Brazil 22460-320
Annals of mathematics, Tome 172 (2010) no. 3, pp. 1879-1906.

Voir la notice de l'article provenant de la source Annals of Mathematics website

We study complete minimal graphs in $\mathbb{H}\times\mathbb{R}$, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon $\Gamma$ in $\mathbb{H}$. We give necessary and sufficient conditions on the “lengths” of the sides of the polygon (and all inscribed polygons in $\Gamma$) that ensure the existence of such a graph. We then apply this to construct entire minimal graphs in $\mathbb{H}\times\mathbb{R}$ that are conformally the complex plane $\mathbb{C}$. The vertical projection of such a graph yields a harmonic diffeomorphism from $\mathbb{C}$ onto $\mathbb{H}$, disproving a conjecture of Rick Schoen and S.-T. Yau.
DOI : 10.4007/annals.2010.172.1879

Pascal Collin 1 ; Harold Rosenberg 2

1 Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne<br/>F-31062 Toulouse Cedex 9<br/>France
2 IMPA<br/>Estrada Dona Castorina 110<br/>Rio de Janeiro<br/>Brazil 22460-320 
@article{10_4007_annals_2010_172_1879,
     author = {Pascal Collin and Harold Rosenberg},
     title = {Construction of harmonic diffeomorphisms  and minimal graphs},
     journal = {Annals of mathematics},
     pages = {1879--1906},
     publisher = {mathdoc},
     volume = {172},
     number = {3},
     year = {2010},
     doi = {10.4007/annals.2010.172.1879},
     mrnumber = {2726102},
     zbl = {1209.53010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.1879/}
}
TY  - JOUR
AU  - Pascal Collin
AU  - Harold Rosenberg
TI  - Construction of harmonic diffeomorphisms  and minimal graphs
JO  - Annals of mathematics
PY  - 2010
SP  - 1879
EP  - 1906
VL  - 172
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.1879/
DO  - 10.4007/annals.2010.172.1879
LA  - en
ID  - 10_4007_annals_2010_172_1879
ER  - 
%0 Journal Article
%A Pascal Collin
%A Harold Rosenberg
%T Construction of harmonic diffeomorphisms  and minimal graphs
%J Annals of mathematics
%D 2010
%P 1879-1906
%V 172
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.1879/
%R 10.4007/annals.2010.172.1879
%G en
%F 10_4007_annals_2010_172_1879
Pascal Collin; Harold Rosenberg. Construction of harmonic diffeomorphisms  and minimal graphs. Annals of mathematics, Tome 172 (2010) no. 3, pp. 1879-1906. doi : 10.4007/annals.2010.172.1879. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.1879/

Cité par Sources :