Special Liouville metric with the Ricci condition

Kenmotsu, Katsuei

doi:10.1090/proc/15652

Special Liouville metric with the Ricci condition
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by Katsuei Kenmotsu PDF

Proc. Amer. Math. Soc. 150 (2022), 345-350 Request permission

Abstract:

Two necessary conditions for the induced metrics of parallel mean curvature surfaces in a complex space form of complex two-dimension are observed. One is similar to the Ricci condition of the classical surface theory in Euclidean three-space and the other is related to the Liouville metric. Conversely, we prove that a special type of the Liouville metric on a domain in the Euclidean two-plane whose Gaussian curvature satisfies the differential equation similar to the Ricci condition is explicitly determined by an elliptic function. We have isometric immersions from a simply connected two-dimensional Riemannian manifold with the special type of the Liouville metric satisfying the Ricci condition to the complex hyperbolic plane with parallel mean curvature vector.

References

Shiing Shen Chern and Jon Gordon Wolfson, Minimal surfaces by moving frames, Amer. J. Math. 105 (1983), no. 1, 59–83. MR 692106, DOI 10.2307/2374381
Jost-Hinrich Eschenburg, Irwen Válle Guadalupe, and Renato de Azevedo Tribuzy, The fundamental equations of minimal surfaces in $\textbf {C}\textrm {P}^2$, Math. Ann. 270 (1985), no. 4, 571–598. MR 776173, DOI 10.1007/BF01455305
Jost-Hinrich Eschenburg and Renato de Azevedo Tribuzy, Constant mean curvature surfaces in $4$-space forms, Rend. Sem. Mat. Univ. Padova 79 (1988), 185–202. MR 964030
Shinya Hirakawa, Constant Gaussian curvature surfaces with parallel mean curvature vector in two-dimensional complex space forms, Geom. Dedicata 118 (2006), 229–244. MR 2239458, DOI 10.1007/s10711-005-9038-8
Katsuei Kenmotsu, Parallel mean curvature tori in $\Bbb CP^2$ and $\Bbb CH^2$, Tohoku Math. J. (2) 70 (2018), no. 3, 475–485. MR 3856777, DOI 10.2748/tmj/1537495357
Katsuei Kenmotsu, Complete parallel mean curvature surfaces in two-dimensional complex space-forms, Bull. Braz. Math. Soc. (N.S.) 49 (2018), no. 4, 775–788. MR 3880318, DOI 10.1007/s00574-018-0081-0
Kazuyoshi Kiyohara, Compact Liouville surfaces, J. Math. Soc. Japan 43 (1991), no. 3, 555–591. MR 1111603, DOI 10.2969/jmsj/04330555
Andreas Knauf and Yakov G. Sinai, Classical nonintegrability, quantum chaos, DMV Seminar, vol. 27, Birkhäuser Verlag, Basel, 1997. With a contribution by Viviane Baladi. MR 1461015, DOI 10.1007/978-3-0348-8932-2
H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
Makoto Sakaki, Minimal surfaces with the Ricci condition in $4$-dimensional space forms, Proc. Amer. Math. Soc. 121 (1994), no. 2, 573–577. MR 1195731, DOI 10.1090/S0002-9939-1994-1195731-X
Amalia-Sofia Tsouri and Theodoros Vlachos, Minimal surfaces in spheres and a Ricci-like condition, Manuscripta Math. 166(2021), 561–588, online: 20 November 2020, https://doi.org/10.1007/s00229-020-01254-7.
Theodoros Vlachos, Minimal surfaces in a sphere and the Ricci condition, Ann. Global Anal. Geom. 17 (1999), no. 2, 129–150. MR 1675407, DOI 10.1023/A:1006552201857