Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on 3-manifolds

Journal of Differential Geometry

Volumn 33, Issue 2, 1991, Pages 413-443

  White, Brian ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84972519620     PISSN: 0022040X     EISSN: 1945743X     Source Type: Journal    
DOI: 10.4310/jdg/1214446325     Document Type: Article

References (37)

1. F. J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. (2) 87 (1968) 321-391.
2. F. J. Almgren, R. Schoen & L. Simon, Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals, Acta Math. 139 (1977) 217-265.
3. F. J. Almgren & L. Simon, Existence of embedded solutions of Plateau \s problem, Ann. Scuola Norm. Sup. Pisa 6 (1979) 447-495.
4. L. Cesari, An existence theorem of calculus of variations for integrals parametric surfaces, Amer. J. Math. 74 (1952) 265-295.
5. J. M. Danskin, On the existence of minimizing surfaces in parametric double integral problems of the calculus of variations, Rev. Mat. Univ. Parma 3 (1952) 43-63.
6. J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 302-306.
7. J. Douglas, Minimal surfaces of higher topological structure, Ann. of Math. (2) 40 (1939) 205-298.
8. 3 are planes, Bull. Amer. Math. Soc. (N. S.) 1 (1979) 903-906.
9. H. Federer, Geometric measure theory, Springer, Berlin, 1969.
10. H. Federer & W. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960) 458-520.
11. D. Fischer-Colbrie & R. Schoen, The structure of complete stable maximal surfaces in 3-manifolds, Comm. Pure Appl. Math. 33 (1980) 199-211.
12. K. Gerhardt, Regularity of solutions of nonlinear variational inequalities, Arch. Rational Mech. Anal. 52 (1973) 389-393.
13. D. Gilbarg & N. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer, Berlin, 1983.
14. E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, Basel, 1984.
15. R. Gulliver, On the regularity of generalized surfaces of constant mean curvature, Bull. Amer. Math. Soc. 77 (1971) 243-245.
16. R. Gulliver, Regularity of minimizing surfaces of prescribed mean curvature, Ann. of Math. (2) 97 (1973) 275-305.
17. R. Hardt, On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand, Comm. Partial Differential Equations 2 (1977) 1163-1232.
18. J. Hass & P. Scott, Existence of least area surfaces in 3-manifolds, Trans. Amer. Math. Soc. 310 (1988) 87-114.
19. E. Heinz & S. Hildebrandt, Some remarks on minimal surfaces in riemannian manifolds, Comm. Pure Appl. Math. 23 (1970) 371-377.
20. J. Hempel, Three-manifolds, Princeton Univ. Press, Princeton, NJ, 1979.
21. S. Hildebrandt, Boundary behavior of minimal surfaces, Arch. Rational Mech. Anal. 35 (1969) 47-82.
22. F. H. Lin, Regularity for a class of parametric obstacle problems, Ph. D. thesis, Univ. of Minnesota, 1985.
23. F. H. Lin, Existence and regularity of embedded disks, J. Differential Geometry 27 (1988) 553-560.
24. F. H. Lin, Estimates for surfaces which are stationary for an elliptic parametric integral, preprint.
25. W. H. Meeks, III & S. -T. Yau, The classical Plateau problem and the topology of threedimensional manifolds, Topology 21 (1982) 409-442.
26. C. B. Morrey, The problem of Plateau on a riemannian manifold, Ann. of Math. (2) 49 (1948) 807-851.
27. C. B. Morrey, Parametric variational problems for double integrals, Comm. Pure Appl. Math. 14 (1961) 569-575.
28. R. Osserman, A proof of the regularity everywhere of the classical solution to Plateau's problem, Ann. of Math. (2) 91 (1970) 550-569.
29. T. Rado, On the problem of Plateau, Springer, Berlin, 1933; reprint, 1971.
30. R. Schoen & L. Simon, A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functions, Indiana Univ. Math. J. 31 (1982) 415-434.
31. A. G. Sigalov, Two-dimensional problems of the calculus of variations, Uspekhi Mat. Nauk6 (1951) 16-101.
32. F. Tomi & A. J. Tromba, Extreme curves bound embedded minimal surfaces of the type of the disk, Math. Z. 158 (1978) 137-145.
33. B. White, A regularity theorem for minimizing hypersurfaces mod p, Geometric Measure Theory and the Calculus of Variations, Proc. Sympos. Pure Math., Vol. 44, Amer. Math. Soc., 1986, 413-427.
34. B. White, Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math. 88 (1987) 243-256.
35. B. White, The space of m-dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. J. 36 (1987) 567-603.
36. B. White, New applications of mapping degrees to minimal surface theory, J. Differential Geometry 29 (1989) 143-162.
37. B. White, A new proof of the compactness theorem for integral currents, Comment. Math. Helv. 64 (1989) 207-220.