Constant mean curvature surfaces with planar boundary

Duke Mathematical Journal

Volumn 85, Issue 3, 1996, Pages 583-604

  López, Rafael ^a   Montiel, Sebastián ^a  

^a UNIVERSITY OF GRANADA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 16144368017     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-96-08522-1     Document Type: Article

References (35)

1. U. ABRESCH, Constant mean curvature tori in terms of elliptic functions, J. Reine Angew. Math. 394 (1987), 169-192.
2. A. D. ALEXANDROV, Uniqueness theorems for surfaces in the large, V. Vestnik Leningrad Univ. 13: 19 (1958), 5-8.
3. F. J. ALMGREN, Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints, Mem. Amer. Math. Soc. 4., Amer. Math. Soc., Providence, 1976, no. 165.
4. J. L. BARBOSA AND M. P. DO CARMO, A proof of a general isoperimetric inequality for surfaces, Math. Z. 173 (1980), 13-28.
5. 3 in terms of theta functions, Math. Ann. 290 (1991), 209-245.
6. H. BRÉZIS AND J. M. CORON, Multiple solutions of H-systems and Rellich's conjecture, Comm. Pure Appl. Math. 37 (1984), 149-187.
7. S. Y. CHENG, Eigenfunctions and nodal sets, Comm. Math. Helv. 51 (1976), 43-55.
8. R. EARP, F. BRITO, W. H. MEEKS III, AND H. ROSENBERG, Structure theorems for constant mean curvature surfaces bounded by a planar curve, Indiana Univ. Math. J. 40 (1991), 333-343.
9. H. FEDERER, Geometric Measure Theory, Springer-Verlag, New York, 1969.
10. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, 1983.
11. T. L. HEATH, Greek Mathematics, Dover, New York, 1963.
12. H. HEINZ, Über die Existenz einer Flasche konstanter mittlerer Krümmung bei vorgebener Berandung, Math. Ann. 127 (1954), 258-287.
13. _, On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectificable boundary, Arch. Rational Mech. Anal. 35 (1969), 249-252.
14. S. HILDEBRANT, On the Plateau problem for surfaces of constant mean curvature, Comm. Pure Appl. Math. 23 (1970), 97-114.
15. D. HOFFMAN AND H. ROSENBERG, Surfaces minimales et solutions de problämes variationnels, Soc. Math. France, Journée annuelle du 5 Juin 1993.
16. E. HOPF, Elementare Bermekungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung von Elliptischen Typen, Preuss. Akad. Wiss. 19 (1927), 147-152.
17. _, Differential Geometry in the Large, Lecture Notes in Math. 1000, Springer-Verlag, Berlin, 1983.
18. N. KAPOULEAS, Compact constant mean curvature surfaces in Euclidean three-space, J. Differential Geom. 33 (1991), 683-715.
19. N. KOREVAAR AND R. KUSNER, The global structure of constant mean curvature surfaces, Invent. Math. 114 (1993), 311-332.
20. N. KOREVAAR, R. KUSNER, AND B. SOLOMON, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), 465-503.
21. R. KUSNER, Global geometry of extremal surfaces in three-space, Ph.D. thesis, University of California, Berkeley, 1988.
22. R. LÓPEZ AND S. MONTIEL, Constant mean curvature discs with bounded area, Proc. Amer. Math. Soc. 123 (1995), 1555-1558.
23. S. MONTIEL AND A. ROS, "Compact hypersurfaces: The Alexandrov theorem for higher-order mean curvatures" in Differential Geometry, ed. by B. Lawson and K. Tenenblat, Pitman Monographs Surveys Pure Appl. Math. 52, Longman, Harlow, Great Britain, 1991, 279-296.
24. U. PINKALL AND I. STERLING, On the classification of constant mean curvature tori, Ann. of Math. (2) 130 (1989), 407-451.
25. A. ROS AND H. ROSENBERG, to appear in J. Differential Geom.
26. J. SERRIN, The problem of Dirichlet for quasilinear elliptic equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413-496.
27. _, On surfaces of constant mean curvature which span a given space curve, Math. Z. 112 (1969), 77-88.
28. K. STEFFEN, On the existence of surfaces with prescribed mean curvature and boundary, Math. Z. 146 (1976), 113-135.
29. K. STEFFEN AND H. C. WENTE, The nonexistence of branch points in a solution to certain classes of Plateau type variational problems, Math. Z. 163 (1978), 211-238.
30. M. STRUWE, Plateau's Problem and the Calculus of Variations, Math. Notes 35, Princeton University Press, Princeton, 1988.
31. H. C. WENTE, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318-344.
32. _, A general existence theorem for surfaces of constant mean curvature, Math. Z. 120 (1971), 277-288.
33. _, Large solutions to the volume constrained Plateau problem, Arch. Rational Mech. Anal. 75 (1980), 59-77.
34. _, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), 193-243.
35. H. WERNER, Problem von Douglas für Flächen konstanter mittlerer Krümmung, Math. Ann. 135 (1957), 303-319.