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Constant mean curvature surfaces constructed by fusing Wente tori. (English) Zbl 0840.53005
This paper contains detailed proofs of the results announced in [the author, Proc. Natl. Acad. Sci. USA 89, No. 12, 5695-5698 (1992; Zbl 0763.53014)]. The main geometric result is the existence of infinitely many immersed \(C^\infty\) surfaces of constant mean curvature \(H = 1\) and genus \(g \geq 2\) into Euclidean 3-space. The main difficulties (as compared to the case of fusing Delaunay pieces) result from the different geometry of the flat point set. This requires completely new balancing concepts. The paper is conceptually difficult and technically extremely involved, as is emphasized by an index of more than fifty notational symbols given in the introduction.
Reviewer: D.Ferus (Berlin)

MSC:
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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[1] [Ab1] U. Abresch: Constant mean curvature tori in terms of elliptic functions. J. Reine Angew. Math.374, 169-192 (1987) · Zbl 0597.53003
[2] [Ab2] U. Abresch: Old and new periodic solutions of the sinh-Gordon equation. Seminar on new results in non-linear partial differential equations. Braunschweig: Vieweg (1987)
[3] [A] A. D. Alexandrov: Uniqueness theorems for surfaces in the largeI, Vestnik Leningrad Univ. Math.11, 5-17 (1956)
[4] [Bo] A. I. Bobenko: All constant mean curvature tori in ?3, \(\mathbb{S}^3 \) , ?3 in terms of theta-functions. Math. Ann.290, 209-245 (1991) · Zbl 0711.53007
[5] [B-dC] L. Barbosa, M. doCarmo: Stability of hypersurfaces with constant mean curvature. Math. Z.185, 339-353 (1984) · Zbl 0529.53006
[6] [Ch] I. Chavel: Eigenvalues in Riemannian Geometry. London: Academic Press, 1984
[7] [C] S. S. Chern: Some new characterizations of the Euclidean sphere. Duke Math. J.12, 279-290 (1945) · Zbl 0063.00833
[8] [D] D. Delaunay: Sur la surface de revolution dont la courbure moyenne est constante. J. Math. Pures Appl.6, 309-320 (1841)
[9] J. Dorfmeister, H. Wu: Constant mean curvature surfaces and loop groups (Preprint) · Zbl 0779.53004
[10] [E-K-T] N. Ercolani, H. Knorrer, E. Trubowitz: Hyperelliptic curves that generate constant mean curvature tori inR 3. The Verdier Memorial Conference in Integrable Systems: Actes du colloque international de luminy (1991), ed. by Olivier Babelon, Pierre Cartier and Yvette Kosmann-Schwarzbach, Basel: Birkhäuser
[11] [G-T] D. Gilbarg, N. S. Trudinger: Elliptic partial differential equations of second order. 2nd Edition, Berlin Heidelberg New York: Springer, 1983 · Zbl 0562.35001
[12] H. Hopf: Lectures on Differential Geometry in the Large. Notes by John Gray, Stanford University · Zbl 0669.53001
[13] [HS] Wu-Yi Hsiang: Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces, I. J. Differ. Geom.17, 337-356 (1982) · Zbl 0493.53043
[14] [J] J. H. Jellet: Sur la surface dont la coubure moyenne est constante. J. Math. Pures Appl.18, 163-167 (1853)
[15] [K1] N. Kapouleas: Constant mean curvature surfaces in Euclidean three-space. Bull. AMS17, 318-320 (1987) · Zbl 0636.53010
[16] [K2] N. Kapouleas: Complete constant mean curvature surfaces in Euclidean three-space. Ann. Math. (2)131, 239-330 (1990) · Zbl 0699.53007
[17] [K3] N. Kapouleas: Compact constant mean curvature surfaces in Euclidean three-space. J. Differ. Geom.33, 683-715 (1991) · Zbl 0727.53063
[18] [K4] N. Kapouleas: Constant mean curvature surfaces constructed by fusing Wente tori. Proc. Nat. Acad. Sci. USA89, 5695-5698 (1992) · Zbl 0763.53014
[19] [K-K-S] N. Korevaar, R. Kusner, B. Solomon: The structure of complete embedded surfaces with constant mean curvature. J. Diff. Geom.30, 465-503 (1989) · Zbl 0726.53007
[20] [P-S] U. Pinkall, I. Sterling: On the classification of constant mean curvature tori. Ann. Math.130, 407-451 (1989) · Zbl 0683.53053
[21] [Tr] F. Tricomi, M. Krafft Elliptische Functionen. Leipzig: Akad, Verl. Ges. Geest. u. Portig, 1948
[22] M. Timmreck, U. Pinkall, D. Ferus: Constant mean curvature planes with inner rotational symmetry in Euclidean 3-space (Preprint) · Zbl 0799.53011
[23] [Wa] R. Walter: Explicit examples to the H-Problem of Heinz Hopf. Geom. Dedicata23, 187-213 (1987) · Zbl 0615.53050
[24] [W1] H. C. Wente: Counterexample to a conjecture of H. Hopf. Pac. J. Math.121, 193-243 (1986) · Zbl 0586.53003
[25] [W2] H. C. Wente: Twisted tori of constant mean curvature inR 3, Seminar on new results in non-linear partial differential equations. Braunschweig: Vieweg 1987
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