Abstract
We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.
A Derivative of the monodromy
The following proposition is adapted from [23, Proposition 9].
Proposition 8.
Let
Proof.
Since
Following the method of variation of constants , the function
Hence (writing
We have by definition
Hence since
B Smoothness of maps between Banach spaces
The following proposition is useful to prove that the maps considered in
this paper are smooth maps between Banach spaces.
The Banach algebra
Proposition 9.
Let
where we identify
Then
Proof.
We expand f in Laurent series with respect to λ
and power series with respect to
For any
Let
Since
Hence by inequality (B.1),
series (B.2) converges normally, so
Acknowledgements
I would like to thank the referee for several interesting suggestions (see Remark 4 and Section 10).
References
[1] S. B. Chae, Holomorphy and calculus in normed spaces, Monogr. Textb. Pure Appl. Math. 92, Marcel Dekker, New York 1985. Search in Google Scholar
[2] J. Dorfmeister and G. Haak, On constant mean curvature surfaces with periodic metric, Pacific J. Math. 182 (1998), no. 2, 229–287. 10.2140/pjm.1998.182.229Search in Google Scholar
[3] J. Dorfmeister, F. Pedit and H. Wu, Weierstrass-type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), no. 4, 633–668. 10.4310/CAG.1998.v6.n4.a1Search in Google Scholar
[4] J. Dorfmeister and H. Wu, Construction of constant mean curvature n-noids from holomorphic potentials, Math. Z. 258 (2008), no. 4, 773–803. 10.1007/s00209-007-0197-1Search in Google Scholar
[5] O. Forster, Lectures on Riemann surfaces, Grad. Texts in Math. 81, Springer, New York 1981. 10.1007/978-1-4612-5961-9Search in Google Scholar
[6] S. Fujimori, S. Kobayashi and W. Rossman, Loop group methods for constant mean curvature surfaces, preprint (2006), https://arxiv.org/abs/math/0602570. Search in Google Scholar
[7] A. Gerding, F. Pedit and N. Schmitt, Constant mean curvature surfaces: An integrable systems perspective, Harmonic maps and differential geometry, Contemp. Math. 542, American Mathematical Society, Providence (2011), 7–39. 10.1090/conm/542/10697Search in Google Scholar
[8] K. Grosse Brauckmann, R. B. Kusner and J. M. Sullivan, Triunduloids: Embedded constant mean curvature surfaces with three ends and genus zero, J. reine angew. Math. 564 (2003), 35–61. 10.1515/crll.2003.093Search in Google Scholar
[9] K. Grosse-Brauckmann, R. B. Kusner and J. M. Sullivan, Coplanar constant mean curvature surfaces, Comm. Anal. Geom. 15 (2007), no. 5, 985–1023. 10.4310/CAG.2007.v15.n5.a4Search in Google Scholar
[10] L. Heller, S. Heller and N. Schmitt, Navigating the space of symmetric CMC surfaces, J. Differential Geom. 110 (2018), no. 3, 413–455. 10.4310/jdg/1542423626Search in Google Scholar
[11]
S. Heller,
Higher genus minimal surfaces in
[12] S. Heller, Lawson’s genus two surface and meromorphic connections, Math. Z. 274 (2013), no. 3–4, 745–760. 10.1007/s00209-012-1094-9Search in Google Scholar
[13] S. Heller, A spectral curve approach to Lawson symmetric CMC surfaces of genus 2, Math. Ann. 360 (2014), no. 3–4, 607–652. 10.1007/s00208-014-1044-4Search in Google Scholar
[14] N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239–330. 10.2307/1971494Search in Google Scholar
[15] M. Kilian, S.-P. Kobayashi, W. Rossman and N. Schmitt, Constant mean curvature surfaces of any positive genus, J. Lond. Math. Soc. (2) 72 (2005), no. 1, 258–272. 10.1112/S0024610705006472Search in Google Scholar
[16] M. Kilian, I. McIntosh and N. Schmitt, New constant mean curvature surfaces, Exp. Math. 9 (2000), no. 4, 595–611. 10.1080/10586458.2000.10504663Search in Google Scholar
[17] M. Kilian, W. Rossman and N. Schmitt, Delaunay ends of constant mean curvature surfaces, Compos. Math. 144 (2008), no. 1, 186–220. 10.1112/S0010437X07003119Search in Google Scholar
[18] N. J. Korevaar, R. Kusner and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465–503. 10.4310/jdg/1214443598Search in Google Scholar
[19] T. Raujouan, On Delaunay ends in the DPW method, preprint (2017), https://arxiv.org/abs/1710.00768. 10.1512/iumj.2020.69.8123Search in Google Scholar
[20] N. Schmitt, Constant mean curvature n-noids with platonic symmetries, preprint (2007), https://arxiv.org/abs/math/0702469. Search in Google Scholar
[21] N. Schmitt, M. Kilian, S.-P. Kobayashi and W. Rossman, Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms, J. Lond. Math. Soc. (2) 75 (2007), no. 3, 563–581. 10.1112/jlms/jdm005Search in Google Scholar
[22] M. E. Taylor, Introduction to differential equations, Pure Appl. Undergrad. Texts 14, American Mathematical Society, Providence 2011. Search in Google Scholar
[23] M. Traizet, Opening nodes on horosphere packings, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5701–5725. 10.1090/tran/6550Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston