On harmonic quasiconformal immersions of surfaces in ℝ³

Alarcón, Antonio; López, Francisco

doi:10.1090/S0002-9947-2012-05658-3

On harmonic quasiconformal immersions of surfaces in $\mathbb {R}^3$
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by Antonio Alarcón and Francisco J. López PDF

Trans. Amer. Math. Soc. 365 (2013), 1711-1742 Request permission

Abstract:

This paper is devoted to the study of the global properties of harmonically immersed Riemann surfaces in $\mathbb {R}^3.$ We focus on the geometry of complete harmonic immersions with quasiconformal Gauss map, and in particular, of those with finite total curvature. We pay special attention to the construction of new examples with significant geometry.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787, DOI 10.1090/ulect/038
Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
Tobias H. Colding and William P. Minicozzi II, The Calabi-Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), no. 1, 211–243. MR 2373154, DOI 10.4007/annals.2008.167.211
Pascal Collin, Topologie et courbure des surfaces minimales proprement plongées de $\mathbf R^3$, Ann. of Math. (2) 145 (1997), no. 1, 1–31 (French). MR 1432035, DOI 10.2307/2951822
V. Ya. Gutlyanskiĭ and V. I. Ryazanov, On the theory of the local behavior of quasiconformal mappings, Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995), no. 3, 31–58 (Russian, with Russian summary); English transl., Izv. Math. 59 (1995), no. 3, 471–498. MR 1347077, DOI 10.1070/IM1995v059n03ABEH000021
Erhard Heinz, Über die Lösungen der Minimalflächengleichung, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt. 1952 (1952), 51–56 (German). MR 54182
Frédéric Hélein, Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002. Translated from the 1996 French original; With a foreword by James Eells. MR 1913803, DOI 10.1017/CBO9780511543036
Frédéric Hélein and John C. Wood, Harmonic maps, Handbook of global analysis, Elsevier Sci. B. V., Amsterdam, 2008, pp. 417–491, 1213. MR 2389639, DOI 10.1016/B978-044452833-9.50009-7
Alfred Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13–72. MR 94452, DOI 10.1007/BF02564570
Luquésio P. Jorge and William H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), no. 2, 203–221. MR 683761, DOI 10.1016/0040-9383(83)90032-0
Luquésio P. de M. Jorge and Frederico Xavier, A complete minimal surface in $\textbf {R}^{3}$ between two parallel planes, Ann. of Math. (2) 112 (1980), no. 1, 203–206. MR 584079, DOI 10.2307/1971325
Luquésio P. de M. Jorge and Frederico V. Xavier, An inequality between the exterior diameter and the mean curvature of bounded immersions, Math. Z. 178 (1981), no. 1, 77–82. MR 627095, DOI 10.1007/BF01218372
D. Kalaj, Gauss map of a harmonic surface. Preprint (arXiv:1103.1576).
Tilla Klotz, Surfaces harmonically immersed in $E^{3}$, Pacific J. Math. 21 (1967), 79–87. MR 232321
Tilla Klotz, A complete $R_{\Lambda }$-harmonically immersed surface in $E^{3}$ on which $H\not =0$, Proc. Amer. Math. Soc. 19 (1968), 1296–1298. MR 233319, DOI 10.1090/S0002-9939-1968-0233319-X
Tilla Klotz Milnor, Harmonically immersed surfaces, J. Differential Geometry 14 (1979), no. 2, 205–214. MR 587548
Tilla Klotz Milnor, Mapping surfaces harmonically into $E^{n}$, Proc. Amer. Math. Soc. 78 (1980), no. 2, 269–275. MR 550511, DOI 10.1090/S0002-9939-1980-0550511-0
Paul Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
Francisco J. López, Some Picard theorems for minimal surfaces, Trans. Amer. Math. Soc. 356 (2004), no. 2, 703–733. MR 2022717, DOI 10.1090/S0002-9947-03-03213-6
Francisco J. López and Antonio Ros, On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991), no. 1, 293–300. MR 1085145
Paul MacManus, Quasiconformal mappings and Ahlfors-David curves, Trans. Amer. Math. Soc. 343 (1994), no. 2, 853–881. MR 1202420, DOI 10.1090/S0002-9947-1994-1202420-6
W.H. Meeks III and J. Pérez, Finite type annular ends for harmonic functions. Preprint (arXiv:0909.1963).
Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330
Robert Osserman, On complete minimal surfaces, Arch. Rational Mech. Anal. 13 (1963), 392–404. MR 151907, DOI 10.1007/BF01262706
Robert Osserman, Global properties of minimal surfaces in $E^{3}$ and $E^{n}$, Ann. of Math. (2) 80 (1964), 340–364. MR 179701, DOI 10.2307/1970396
Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
Joaquín Pérez and Antonio Ros, Some uniqueness and nonexistence theorems for embedded minimal surfaces, Math. Ann. 295 (1993), no. 3, 513–525. MR 1204835, DOI 10.1007/BF01444900
Richard M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), no. 4, 791–809 (1984). MR 730928
Leon Simon, A Hölder estimate for quasiconformal maps between surfaces in Euclidean space, Acta Math. 139 (1977), no. 1-2, 19–51. MR 452746, DOI 10.1007/BF02392233
M. Weber, Construction of Harmonic Surfaces with Prescribed Geometry. Lecture Notes in Computer Science, 2010, Volume 6327/2010, 170–173.
Brian White, Complete surfaces of finite total curvature, J. Differential Geom. 26 (1987), no. 2, 315–326. MR 906393
Frederico Xavier, Convex hulls of complete minimal surfaces, Math. Ann. 269 (1984), no. 2, 179–182. MR 759107, DOI 10.1007/BF01451417
F. Xavier, Embedded, simply connected, minimal surfaces with bounded curvature, Geom. Funct. Anal. 11 (2001), no. 6, 1344–1356. MR 1878323, DOI 10.1007/s00039-001-8233-5