Abstract
A conformal map from a Riemann surface to the Euclidean four-space is explained in terms of its twistor lift. A local factorization of a differential of a conformal map is obtained. As an application, the factorization of a differential provides an upper bound of the area of a super-conformal map around a branch point.
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Bayard P., Lawn M., Roth J.: Spinorial representation of surfaces into 4-dimensional space forms. Ann. Glob. Ana.l Geom. 44(4), 433–453 (2013)
Bohle C., Leschke K., Pedit F., Pinkall U.: Conformal maps from a 2-torus to the 4-sphere. J. Reine Angew. Math. 671, 1–30 (2012)
Bryant R. L.: Conformal and minimal immersions of compact surfaces into the 4-sphere. J. Diff. Geom. 17(3), 455–473 (1982)
Burstall, F. E., Ferus, D., Leschke, K.; Pedit, F., Pinkall, U.: Conformal geometry of surfaces in S 4 and quaternions, Lecture Notes in Mathematics 1772, Springer-Verlag, Berlin (2002).
Castro I., Urbano F.: On twistor harmonic surfaces in the complex projective plane. Math. Proc. Cambridge Philos. Soc. 122(1), 115–129 (1997)
Enneper A.: Analytisch-geometrische Untersuchungen. Z. Math. Phys. 9, 96–125 (1864)
Ferus D., Leschke K., Pedit F., Pinkall U.: Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori. Invent. Math. 146(3), 507–593 (2001)
Friedrich T.: On surfaces in four-spaces. Ann. Global Anal. Geom. 2(3), 257–287 (1984)
Greene, R. E., Krantz, S. G.: Function theory of one complex variable, Graduate Studies in Mathematics 40, 3rd ed., American Mathematical Society, Providence, RI (2006).
Kamberov, G.; Norman, P.; Pedit, F.; Pinkall, U.: Quaternions, Spinor and Surfaces, Contemporary Mathematics (AMS), 299, 2002.
Hopf, H.: Differential geometry in the large, Lecture Notes in Mathematics 1000, Notes taken by Peter Lax and John Gray; With a preface by S. S. Chern, Springer-Verlag, Berlin, 1983.
Leschke, K.; Pedit, F.: Bäcklund transforms of conformal maps into the 4-sphere, PDEs, submanifolds and affine differential geometry, Banach Center Publ. 69, Polish Acad. Sci., Warsaw, 2005, 103–118.
Moriya K.: The denominators of Lagrangian surfaces in complex Euclidean plane. Ann. Global Anal. Geom. 34(1), 1–20 (2008)
Moriya K.: Super-conformal surfaces associated with null complex holomorphic curves. Bull. Lond. Math. Soc. 41(2), 327–331 (2009)
Moriya K.: Darboux Transforms of a Harmonic Inverse Mean Curvature Surface, Geometry, vol. 2013, Article ID 902092, 9 pages (2013).
Moriya K.: A factorization of a super-conformal map. Israel J. Math. 207(1), 331–359 (2015)
Pedit, F., Pinkall, U.: Quaternionic analysis on Riemann surfaces and differential geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., 1998, Extra Vol. II, 389–400 (electronic).
Salamon, S.: Topics in four-dimensional Riemannian geometry, Geometry seminar “Luigi Bianchi”, Pisa, 1982, Lecture Notes in Math. 1022, Springer, Berlin, 1983, 33–124.
Schwarz, H.A.: Gesammelte mathematische Abhandlungen. 2 Bände, 1890, Berlin. Springer. Bd. I. XI u. 338 S., Bd. II. VII u. 370 S. gr \({8^{\circ}}\) (1890).
Weierstrass, K.: Über die Flächen, deren mittlere Krümmung überall gleich null ist, Ber. Akad. Wiss. Berlin, 612–625 (1866).
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This work was supported by JSPS KAKENHI Grant Number 25400063 and JSPS KAKENHI Grant Number 23540081.
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Hasegawa, K., Moriya, K. Twistor Lifts and Factorization for Conformal Maps from a Surface to the Euclidean Four-space. Adv. Appl. Clifford Algebras 27, 1243–1262 (2017). https://doi.org/10.1007/s00006-016-0728-0
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DOI: https://doi.org/10.1007/s00006-016-0728-0