Abstract
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.
-
[A] U. Abresch.
@misc{A,
author={Abresch, U.},
NOTE={personal communication},
} -
[Co-K]
P. Collin and R. Krust, "Le problème de Dirichlet pour l’équation des surfaces minimales sur des domaines non bornés," Bull. Soc. Math. France, vol. 119, iss. 4, pp. 443-462, 1991.@article {Co-K, MRKEY = {1136846},
AUTHOR = {Collin, P. and Krust, R.},
TITLE = {Le problème de {D}irichlet pour l'équation des surfaces minimales sur des domaines non bornés},
JOURNAL = {Bull. Soc. Math. France},
FJOURNAL = {Bulletin de la Société Mathématique de France},
VOLUME = {119},
YEAR = {1991},
NUMBER = {4},
PAGES = {443--462},
ISSN = {0037-9484},
CODEN = {BSMFAA},
MRCLASS = {53A10 (35J65 49Q05)},
MRNUMBER = {92m:53007},
MRREVIEWER = {Jo{ã}o Lucas Marques Barbosa},
URL = {http://www.numdam.org/item?id=BSMF_1991__119_4_443_0},
ZBLNUMBER = {0754.53013},
} -
[He] E. Heinz, "Über die Lösungen der Minimalflächengleichung," Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt., vol. 1952, pp. 51-56, 1952.
@article {He, MRKEY = {0054182},
AUTHOR = {Heinz, Erhard},
TITLE = {Über die {L}ösungen der {M}inimalflächengleichung},
JOURNAL = {Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt.},
VOLUME = {1952},
YEAR = {1952},
PAGES = {51--56},
MRCLASS = {49.0X},
MRNUMBER = {14,885c},
MRREVIEWER = {T. Rad{ó}},
} -
[Hu] A. Huber, "On subharmonic functions and differential geometry in the large," Comment. Math. Helv., vol. 32, pp. 13-72, 1957.
@article {Hu, MRKEY = {0094452},
AUTHOR = {Huber, Alfred},
TITLE = {On subharmonic functions and differential geometry in the large},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {32},
YEAR = {1957},
PAGES = {13--72},
ISSN = {0010-2571},
MRCLASS = {30.00 (31.00)},
MRNUMBER = {20 \#970},
MRREVIEWER = {E. F. Beckenbach},
DOI = {10.1007/BF02564570},
} -
[J-S] H. Jenkins and J. Serrin, "Variational problems of minimal surface type. II. Boundary value problems for the minimal surface equation," Arch. Rational Mech. Anal., vol. 21, pp. 321-342, 1966.
@article {J-S, MRKEY = {0190811},
AUTHOR = {Jenkins, Howard and Serrin, James},
TITLE = {Variational problems of minimal surface type. {II}. {B}oundary value problems for the minimal surface equation},
JOURNAL = {Arch. Rational Mech. Anal.},
FJOURNAL = {Archive for Rational Mechanics and Analysis},
VOLUME = {21},
YEAR = {1966},
PAGES = {321--342},
ISSN = {0003-9527},
MRCLASS = {49.00},
MRNUMBER = {32 \#8221},
DOI = {10.1007/BF00282252},
} -
[M] V. Markovic, "Harmonic diffeomorphisms and conformal distortion of Riemann surfaces," Comm. Anal. Geom., vol. 10, iss. 4, pp. 847-876, 2002.
@article {M, MRKEY = {1925504},
AUTHOR = {Markovic, Vladimir},
TITLE = {Harmonic diffeomorphisms and conformal distortion of {R}iemann surfaces},
JOURNAL = {Comm. Anal. Geom.},
FJOURNAL = {Communications in Analysis and Geometry},
VOLUME = {10},
YEAR = {2002},
NUMBER = {4},
PAGES = {847--876},
ISSN = {1019-8385},
MRCLASS = {58E20 (30C62 30F15 30F45 53C43)},
MRNUMBER = {2003i:58029},
MRREVIEWER = {Yoichi Imayoshi},
ZBLNUMBER = {1024.53044},
} -
[N-R] B. Nelli and H. Rosenberg, "Minimal surfaces in ${\Bbb H}^2\times\Bbb R$," Bull. Braz. Math. Soc., vol. 33, iss. 2, pp. 263-292, 2002.
@article {N-R, MRKEY = {1940353},
AUTHOR = {Nelli, Barbara and Rosenberg, Harold},
TITLE = {Minimal surfaces in {${\Bbb H}\sp 2\times\Bbb R$}},
JOURNAL = {Bull. Braz. Math. Soc.},
FJOURNAL = {Bulletin of the Brazilian Mathematical Society. New Series. Boletim da Sociedade Brasileira de Matemática},
VOLUME = {33},
YEAR = {2002},
NUMBER = {2},
PAGES = {263--292},
ISSN = {1678-7544},
MRCLASS = {53A10},
MRNUMBER = {2004d:53014},
DOI = {10.1007/s005740200013},
ZBLNUMBER = {1038.53011},
} -
[SE] R. Sa Earp, "Parabolic and hyperbolic screw motion surfaces in $\Bbb H^2\times\Bbb R$," J. Aust. Math. Soc., vol. 85, iss. 1, pp. 113-143, 2008.
@article {SE, MRKEY = {2460869},
AUTHOR = {Sa Earp, Ricardo},
TITLE = {Parabolic and hyperbolic screw motion surfaces in {$\Bbb H\sp 2\times\Bbb R$}},
JOURNAL = {J. Aust. Math. Soc.},
FJOURNAL = {Journal of the Australian Mathematical Society},
VOLUME = {85},
YEAR = {2008},
NUMBER = {1},
PAGES = {113--143},
ISSN = {1446-7887},
MRCLASS = {53C42},
MRNUMBER = {2010d:53067},
MRREVIEWER = {Pablo Mira},
DOI = {10.1017/S1446788708000013},
ZBLNUMBER = {1178.53060},
} -
[S] R. M. Schoen, "The role of harmonic mappings in rigidity and deformation problems," in Complex Geometry, New York: Dekker, 1993, vol. 143, pp. 179-200.
@incollection {S, MRKEY = {1201611},
AUTHOR = {Schoen, Richard M.},
TITLE = {The role of harmonic mappings in rigidity and deformation problems},
BOOKTITLE = {Complex Geometry},
VENUE={{O}saka, 1990)},
SERIES = {Lecture Notes in Pure and Appl. Math.},
VOLUME = {143},
PAGES = {179--200},
PUBLISHER = {Dekker},
ADDRESS = {New York},
YEAR = {1993},
MRCLASS = {58E20 (32C17 53C21)},
MRNUMBER = {94g:58055},
ZBLNUMBER = {0806.58013},
} -
[S-Y] R. M. Schoen and S. T. Yau, Lectures on Harmonic Maps, Somerville, MA: International Press, 1997, vol. II.
@book {S-Y, MRKEY = {1474501},
AUTHOR = {Schoen, Richard M. and Yau, S. T.},
TITLE = {Lectures on Harmonic Maps},
SERIES = {Conference Proceedings and Lecture Notes in Geometry and Topology},
VOLUME={II},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {1997},
PAGES = {vi+394},
ISBN = {1-57146-002-0},
MRCLASS = {58E20 (53C42)},
MRNUMBER = {98i:58072},
MRREVIEWER = {John C. Wood},
ZBLNUMBER = {0886.53004},
} -
[V] A. Vasil$’$ev, Moduli of Families of Curves for Conformal and Quasiconformal Mappings, New York: Springer-Verlag, 2002, vol. 1788.
@book {V, MRKEY = {1929066},
AUTHOR = {Vasil$'$ev, Alexander},
TITLE = {Moduli of Families of Curves for Conformal and Quasiconformal Mappings},
SERIES = {Lecture Notes in Math.},
VOLUME = {1788},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2002},
PAGES = {x+211},
ISBN = {3-540-43846-7},
MRCLASS = {30-02 (30C35 30C55 30C62 30C75 30F10 30F60)},
MRNUMBER = {2003j:30003},
MRREVIEWER = {Vladimir N. Dubinin},
ZBLNUMBER={0999.30001},
}
Full Article