Abstract
In this chapter we consider the following configuration: a Riemannian manifold X of bounded geometry, some closed Jordan curves Γ j , and a supporting surface ∂K, disjoint from the Γ j . We further assume that the Γ j are contained in a suitable barrier ∂A of nonnegative mean curvature (cf. §2 for details).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Almgren Jr. and L. Simon, Existence of embedded solutions of Plateau’s problem, Ann. Sc. N. Pisa (iv), 6 (1979), 447–495.
M. Freedman, J. Hass, and P. Scott, Least area incompressible surfaces in 3-manifolds, Inv. Math., 71 (1983), 609–642.
M. Grüter and J. Jost, Allard type regularity results for manifolds with free boundaries, Ann. Sci. Norm. Sup. Pisa. In press (1986).
E. Heinz and S. Hildebrandt, Remarks on minimal surfaces in Riemannian manifolds, CPAM 23 (1970), 371–377.
J. Jost, Harmonic maps between surfaces, Lecture Notes in Mathematics, Springer-Verlag, New York, 1062 (1984).
J. Jost, Conformal mappings and the Plateau-Dougles problem, J. Reine Angew. Math., 359 (1985), 37–54.
J. Jost, Existence results for embedded minimal surfaces of controlled topological type I, Ann. Sci. Norm. Sup. Pisa. In press (1986).
J. Jost, On the regularity of minimal surfaces with free boundaries in Riemannian manifolds, Man. Math. (to appear).
J. Jost and H. Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Man. Math., 40 (1982), 27–77.
W. Meeks III, L. Simon, and S.T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. Math., 116 (1982), 621–659.
W. Meeks III and S.T. Yau, The classical Plateau problem and the topology of three-dimensional manifolds, Top. 21 (1982), 409–442.
W. Meeks III and S.T. Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z., 179 (1982), 151–168.
D. Mumford, A remark on Mahler’s compactness theorem, Proc. AMS, 28 (1971), 289–294.
R. Schoen and S.T. Yau, Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with nonnegative scalar curvature, Ann. Math., 110 (1979), 127–142.
K. Sehüffler, Indextheorie für Minimalflächen vom Geschlecht 1 (to appear).
K. Strebel, Ein Klassifizierungsproblem für Riemannsche Flächen vom Geschlecht 1 (to appear).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Jost, J. (1987). On the Existence of Embedded Minimal Surfaces of Higher Genus with Free Boundaries in Riemannian Manifolds. In: Concus, P., Finn, R. (eds) Variational Methods for Free Surface Interfaces. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4656-5_7
Download citation
DOI: https://doi.org/10.1007/978-1-4612-4656-5_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-9101-5
Online ISBN: 978-1-4612-4656-5
eBook Packages: Springer Book Archive