The existence of least area surfaces in 3-manifolds

Hass, Joel; Scott, Peter

doi:10.1090/S0002-9947-1988-0965747-6

The existence of least area surfaces in $3$-manifolds
HTML articles powered by AMS MathViewer

by Joel Hass and Peter Scott PDF

Trans. Amer. Math. Soc. 310 (1988), 87-114 Request permission

Abstract:

This paper presents a new and unified approach to the existence theorems for least area surfaces in $3$-manifolds.

References

Frederick J. Almgren Jr. and Leon Simon, Existence of embedded solutions of Plateau’s problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), no. 3, 447–495. MR 553794
R. Courant, Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, Interscience Publishers, Inc., New York, N.Y., 1950. Appendix by M. Schiffer. MR 0036317
Shiing-shen Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc. 6 (1955), 771–782. MR 74856, DOI 10.1090/S0002-9939-1955-0074856-1

Solution of the problem of Plateau

34

Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI 10.1007/BF02095997
Robert D. Gulliver II, Regularity of minimizing surfaces of prescribed mean curvature, Ann. of Math. (2) 97 (1973), 275–305. MR 317188, DOI 10.2307/1970848

The existence of minimal surfaces in

manifolds

John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
Stefan Hildebrandt, Boundary behavior of minimal surfaces, Arch. Rational Mech. Anal. 35 (1969), 47–82. MR 248650, DOI 10.1007/BF00248494
William Jaco, Finitely presented subgroups of three-manifold groups, Invent. Math. 13 (1971), 335–346. MR 300279, DOI 10.1007/BF01406083

Lectures on minimal submanifolds

Luc Lemaire, Boundary value problems for harmonic and minimal maps of surfaces into manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), no. 1, 91–103. MR 664104
William H. Meeks III and Shing Tung Yau, The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn’s lemma, Topology 21 (1982), no. 4, 409–442. MR 670745, DOI 10.1016/0040-9383(82)90021-0
William H. Meeks III and Shing Tung Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980), no. 3, 441–484. MR 595203, DOI 10.2307/1971088
William W. Meeks III and Shing Tung Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), no. 2, 151–168. MR 645492, DOI 10.1007/BF01214308
William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. MR 678484, DOI 10.2307/2007026
Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807–851. MR 27137, DOI 10.2307/1969401
Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
Robert Osserman, A proof of the regularity everywhere of the classical solution to Plateau’s problem, Ann. of Math. (2) 91 (1970), 550–569. MR 266070, DOI 10.2307/1970637
C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1–26. MR 90053, DOI 10.2307/1970113
Tibor Radó, On Plateau’s problem, Ann. of Math. (2) 31 (1930), no. 3, 457–469. MR 1502955, DOI 10.2307/1968237
J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI 10.2307/1971131

Minimal immersions of closed Riemann surfaces

Richard Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111–126. MR 795231
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
G. P. Scott, On sufficiently large $3$-manifolds, Quart. J. Math. Oxford Ser. (2) 23 (1972), 159–172; correction, ibid. (2) 24 (1973), 527–529. MR 383414, DOI 10.1093/qmath/23.2.159
J. H. C. Whitehead, On $2$-spheres in $3$-manifolds, Bull. Amer. Math. Soc. 64 (1958), 161–166. MR 103473, DOI 10.1090/S0002-9904-1958-10193-7