This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
[All] Allard, W.K.: On the first variation of a varifold. Ann. Math. 95 (1972), 417–491.
[Alm] Almgren, F.J.: Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35 (1986), 451–547.
[AT] Almgren, F.J., Thurston, W.P.: Examples of unknotted curves which bound only surfaces of high genus within their convex hull. Ann. Math. 105(1977), 527–538.
[Al1] Alt, H.W.: Verzweigungspunkte von H-Flächen, I. Math. Z. 127 (1972), 333–362.
[Al2] Alt, H.W.: Verzweigungspunkte von H-Flächen, II. Math. Ann. 201 (1973), 33–55.
[Ba] Barbosa, J.L.: Constant mean curvature surfaces bounded by a planar curve. Matematica Contemporanea 1 (1991), 3–15.
[BJ] Barbosa, J.L., Jorge, L.P.: Stable H-surfaces whose boundary is S 1(1). An. Acad. Bras. Ci. 66 (1994), 259–263.
[Be] Bethuel, F.: Un résultat de régularité pour les solutions de l'équation des surfaces à courbure moyenne prescrite. C.R. Acad. Sci. Paris 314 (1992), 1003–1007.
[BG] Bethuel, F., Ghidaglia, J.M.: Improved regularity of solutions to elliptic equations involving Jacobians and applications. J. Math. Pures et Appliquées 72 (1993), 441–474.
[BR] Bethuel, F., Rey, O.: Multiple solutions to the Plateau problem for nonconstant mean curvature. Duke Math. J. 73 (1994), 593–646.
[BC] Brézis, H.R., Coron, M.: Multiple solutions of H-systems and Rellich's conjecture. Commun. Pure Appl. Math. 37 (1984), 149–187.
[BE] Brito, F., Earp, R.: Geometric configurations of constant mean curvature surfaces with planar boundary. An. Acad. Bras. Ci. 63 (1991), 5–19.
[BZ] Burago, Y.D., Zalgaller, V.A.: Geometric inequalities. Springer-Verlag, New York Heidelberg Berlin, 1988.
[Cr] Croke C.B.: A sharp four dimensional isoperimetric inequality. Comment. Math. Helvetici 59 (1984), 187–192.
[DcG] De Giorgi, E.: Sulla proprietà isoperimetrica dell' ipersfera, nelle classe degli insiemi avanti frontiera orientata di misura finita. Atti. Accad. Naz. Lincei, ser 1, 5 (1958), 33–44.
[Di1] Dierkes, U.: Plateau's problem for surfaces of prescribed mean curature in given regions. Manuscr. Math. 56 (1986), 313–331.
[Di2] Dierkes, U.: A geometric maximum principle for surfaces of prescribed mean curvature in Riemannian manifolds. Z. Anal. Anwend. 8 (2) (1989), 97–102.
[DHKW] Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.: Minimal surfaces vol. 1, vol. 2. Grundlehren math. Wiss. 295, 296. Springer-Verlag, Berlin Heidelberg New York, 1992.
[Du1] Duzaar, F.: Variational inequalities and harmonic mappings. J. Reine Angew. Math. 374 (1987), 39–60.
[Du2] Duzaar, F.: On the existence of surfaces with prescribed mean curvature and boundary in higher dimensions. Ann. Inst. Henri Poincaré (Anal. Non Lineaire) 10 (1993), 191–214.
[Du3] Duzaar, F.: Hypersurfaces with constant mean curvature and prescribed area. Manuscr. Math. 91 (1996), 303–315.
[Du4] Duzaar, F.: Boundary regularity for area minimizing currents with prescribed volume. To appear in J. Geometric Analysis (1988?).
[DF1] Duzaar, F., Fuchs, M.: On the existence of integral currents with prescribed mean curvature vector. Manuscr. Math. 67 (1990), 41–67.
[DF2] Duzaar, F., Fuchs, M.: A general existence theorem for integral currents with prescribed mean curvature form. Bolletino U.M.I. (7) 6-B (1992), 901–912.
[DS1] Duzaar, F., Steffen, K.: Area minimizing hypersurfaces with prescribed volume and boundary. Math. Z. 209 (1992), 581–618.
[DS2] Duzaar, F., Steffen, K.: λ minimizing currents. Manuscr. Mat. 80 (1993), 403–447.
[DS3] Duzaar, F., Steffen, K.: Boundary regularity for minimizing currents with prescribed mean curvature. Calc. Var. 1 (1993), 355–406.
[DS4] Duzaar, F., Steffen, K.: Existence of hypersurfaces with prescribed mean curvature in Riemannian mannifolds. Indiana Univ. Math. J. 45 (1996), 1045–1093.
[DS5] Duzaar, F., Steffen, K.: The Plateau problem for parametric surfaces with prescribed mean curvature. Geometric analysis and the calculus of variations (dedicated to S. Hildebrandt, ed. J. Jost), 13–70, International Press, Cambridge MA, 1996.
[DS6] Duzaar, F., Steffen, K.: Parametric surfaces of least H-energy in a Riemannian manifold. Preprint No. 284, SFB 288 Differential Geometry and Quantum Physics, TU Berlin, 1997.
[EBMR] Earp, R., Brito, F., Meeks III, W.H., Rosenberg, H.: Structure theorems for constant mean curvature surfaces bounded by a planar curve. Indiana Univ. Math. J. 40 (1991), 333–343.
[EL] Eells, J., Lemaire, L.: A report on harmonic maps. Bull. London Math. Soc. 10 (1978), 1–68. Another report on harmonic maps. Bull. London Math. Soc. 20 (1988), 385–542.
[EG] Evans, L.C., Gariepy, L.F.: Measure theory and fine properties of functions. CRC Press, Boca Raton Ann Arbor London, 1992.
[Fe] Federer, H.: Geometric measure theory. Springer-Verlag, Berlin Heidelberg New York, 1969.
[Grü1] Grüter, M.: Regularity of weak H-surfaces. J. Reine Angew. Math. 329 (1981), 1–15.
[Grü2] Grüter, M.: Eine Bemerkung zur Regularität stationärer Punkte von konform invarianten Variationsintegralen. Manuscr. Math. 55 (1986), 451–453.
[Gü] Günther, P.: Einige Vergleichssätze über das Volumenelement eines Riemannschen Raumes. Publ. Math. Debrecen 7 (1960), 258–287.
[Gu1] Gulliver, R.: The Plateau problem for surfaces of prescribed mean curvature in a Riemannian manifold. J. Differ. Geom. 8 (1973), 317–330.
[Gu2] Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. Math. 97 (1973), 275–305.
[Gu3] Gulliver, R.: On the non-existence of a hypersurface of prescribed mean curvature with a given boundary. Manuscr. Math. 11 (1974), 15–39.
[Gu4] Gulliver, R.: Necessary conditions for submanifolds and currents with prescribed mean curvature vector. Seminar on minimal submanifolds, ed. E. Bombieri, Princeton, 1983.
[Gu5] Gulliver, R.: Branched immersions of surfaces and reduction of topological type. I. Math. Z. 145 (1975), 267–288.
[Gu6] Gulliver, R.: Branched immersions of surfaces and reduction of topological type. II. Math. Ann. 230 (1977), 25–48.
[Gu7] Gulliver, R.: A minimal surface with an atypical boundary branch point. Differential Geometry, 211–228, Pitman Monographs Surveys Pure Appl. Math. 52, Longman Sci. Tech., Harlow, 1991.
[GL] Gulliver, R., Lesley, F.D.: On boundary branch points of minimizing surfaces. Arch. Ration. Mech. Anal. 52 (1973), 20–25.
[GOR] Gulliver, R., Osserman, R., Royden, H.L.: A theory of branched immersions of surfaces. Am. J. Math. 95 (1973), 750–812.
[GS1] Gulliver, R., Spruck, J.: The Plateau problem for surfaces of prescribed mean curvature in a cylinder. Invent. Math. 13 (1971), 169–178.
[GS2] Gulliver, R., Spruck, J.: Surfaces of constant mean curvature which have a simple projection. Math. Z. 129 (1972), 95–107.
[GS3] Gulliver, R., Spruck, J.: Existence theorems for parametric surfaces of prescribed mean curvature. Indiana Univ. Math. J. 22 (1972), 445–472.
[HS] Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. Math. 110 (1979), 439–486.
[HW] Hartmann, P., Winter, A.: On the local behaviour of solutions of nonparabolic partial differential equations. Amer. J. Math. 75 (1953), 449–476.
[He1] Heinz, E.: Über die Existenz einer Fläche konstanter mittlerer Krümmung mit gegebener Berandung. Math. Ann. 127 (1954), 258–287.
[He2] Heinz, E.: On the non-existence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch. Rat. Mech. Anal. 35 (1969), 249–252.
[He3] Heinz, E.: Ein Regularitätssatz für Flächen beschränkter mittlerer Krümmung. Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. (1969), 107–118.
[He4] Heinz, E.: Über das Randverhalten quasilinearer elliptischer Systeme mit isothermen Parametern. Math. Z. 113 (1970), 99–105.
[He5] Heinz, E.: Unstable surfaces of constant mean curvature. Arch. Ration. Mech. Anal. 38 (1970), 257–267.
[He6] Heinz, E.: Ein Regularitätssatz für schwache Lösungen nichtlinearer elliptischer Systeme. Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. (1975), 1–13.
[He7] Heinz. E.: Über die Regularität schwacher Lösungen nichtlinarer elliptischer Systeme. Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. (1985), 1–15.
[HH1] Heinz, E., Hildebrandt, S.: Some remerks on minimal surfaces in Riemannian manifolds. Commun. Pure Appl. Math. 23 (1970), 371–377.
[HH2] Heinz, E., Hildebrandt, S.: On the number of branch points of surfaces of bounded mean curvature. J. Differ. Geom. 4 (1970), 227–235.
[HT] Heinz, E., Tomi, F.: Zu einem Satz von S. Hildebrandt über das Randverhalten von Minimalflächen. Math. Z. 111 (1969), 372–386.
[Hi1] Hildebrandt, S.: Boundary behavior of minimal surfaces. Arch. Ration. Mech. Anal. 35 (1969), 47–82.
[Hi2] Hildebrandt, S.: Über Flächen konstanter mittlerer Krümmung. Math. Z. 112 (1969), 107–144.
[Hi3] Hildebrandt, S.: On the Plateau problem for surfaces of prescribed mean curvature. Commun. Pure Appl. Math. 23 (1970), 97–114.
[Hi4] Hildebrandt, S.: Randwertprobleme für Flächen mit vorgeschriebener mittlerer Krümmung und Anwendungen auf die Kapillaritätstheorie I. Math. Z. 112 (1969), 205–213.
[Hi5] Hildebrandt, S.: Über einen neuen Existenzsatz für Flächen vorgeschriebener mittlerer Krümmung. Math. Z. 119 (1971), 267–272.
[Hi6] Hildebrandt, S.: Einige Bemerkungen über Flächen beschränkter mittlerer Krümmung. Math. Z. 115 (1970), 169–178.
[Hi7] Hildebrandt, S.: Maximum principles for minimal surfaces and for surfaces of continuous mean curvature. Math. Z. 128 (1972), 253–269.
[Hi8] Hildebrandt, S.: On the regularity of solutions of two-dimensional variational problems with obstructions. Commun. Pure Appl. Math. 25 (1972), 479–496.
[Hi9] Hildebrandt, S.: Interior C 1+α-regularity of solutions of two-dimensional variational problems with obstacles. Math. Z. 131 (1973), 233–240.
[HK] Hildebrandt, S., Kaul, H.: Two-dimensional variational problems with obstructions, and Plateau's problem for H-surfaces in a Riemannian manifold. Commun. Pure Appl. Math. 25 (1972), 187–223.
[Jä] Jäger, W.: Das Randverhalten von Flächen beschränkter mittlerer Krümmung bei C 1,α-Rändern. Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. (1977), 45–54.
[Jo1] Jost, J.: Lectures on harmonic maps (with applications to conformal mappings and minimal surfaces). Lect. Notes Math. 1161, Springer-Verlag, Berlin Heidelberg New York (1985), 118–192.
[Jo2] Jost, J.: Two-dimensional geometric variational problems. Wiley-Interscience, Chichester New York, 1991.
[Kap] Kapouleas, N.: Compact constant mean curvature surfaces in Euclidean three-space. J. Differ. Geom. 33 (1991), 683–715.
[Kau] Kaul, H.: Ein Einschließungssatz für H-Flächen in Riemannschen Mannigfaltigkeiten. Manuscr. Math. 5 (1971), 103–112.
[Kl] Kleiner, B.: An isoperimetric comparison theorem. Invent. Math. 108 (1992), 37–47.
[LM] López, S., Montiel, S.: Constant mean curvature discs with bounded area. Proc. Amer. Math. Soc. 123 (1995), 1555–1558.
[MM] Massari, U., Miranda, M.: Minimal surfaces of codimension one. North-Holland Mathematical Studies 91, Amsterdam New York Oxford, 1984.
[Ni1] Nitsche, J.C.C.: Vorlesungen über Minimalflächen. Grundlehren math. Wiss., vol. 199. Springer-Verlag, Berlin Heidelberg New York, 1975.
[Ni2] Nitsche, J.C.C.: Lectures on minimal surfaces, vol. 1: Introduction, fundamentals, geometry and basic boundary problems. Cambridge Univ. Press, 1989.
[Os] Osserman, R.: A proof of the regularity everywhere of the classical solution to Plateau's problem. Ann. Math. 91 (1970), 550–569.
[Sch] Schmidt, E.: Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionszahl. Math. Z. 49 (1943/44), 1–109.
[ST] Schüffler, K., Tomi, F.: Ein Indexsatz für Flächen konstanter mittlerer Krümmung. Math. Z. 182 (1983), 245–258.
[Se] Serrin J.: The problem of Dirichlet for quasilinear elliptic differential equations in many independent variables. Phil. Trans. Royal Soc. London 264 (1969), 413–419.
[Si] Simon, L.: Lectures on geometric measure theory. Proc. CMA, Vol. 3, ANU Canberra, 1983.
[Ste1] Steffen, K.: Flächen konstanter mittlerer Krümmung mit vorgegebenem Volumen oder Flächeninhalt. Arch. Ration. Mech. Anal. 49 (1972), 99–128.
[Ste2] Steffen, K.: Ein verbesserter Existenzsatz für Flächen konstanter mittlerer Krümmung. Manuscr. Math. 6 (1972), 105–139.
[Ste3] Steffen, K.: Isoperimetric inequalities and the problem of Plateau. Math. Ann. 222 (1976), 97–144.
[Ste4] Steffen, K.: On the existence of surfaces with prescribed mean curvature and boundary. Math. Z. 146 (1976), 113–135.
[Ste5] Steffen, K.: On the nonuniqueness of surfaces with prescribed constant mean curvature spanning a given contour. Arch. Ration. Mech. Anal. 94 (1986), 101–122.
[SW] Steffen, K., Wente, H.: The non-existence of branch points in solutions to certain classes of Plateau type variational problems. Math. Z. 163 (1978), 211–238.
[Strö] Ströhmer, G.: Instabile Flächen vorgeschriebener mittlerer Krümmung. Math. Z. 174 (1980), 119–133.
[Str1] Struwe, M.: Nonuniqueness in the Plateau problem for surfaces of constant mean curvature. Arch. Ration. Mech. Anal. 93 (1986), 135–157.
[Str2] Struwe, M.: Large H-surfaces via the mountain-pass-lemma. Math. Ann. 270 (1985), 441–459.
[Str3] Struwe, M.: Plateau's problem and the calculus of variations. Mathematical Notes 35, Princeton University Press, Princeton, New Jersey, 1988.
[Str4] Struwe, M.: Multiple solutions to the Dirichlet problem for the equation of prescribed mean curvature. Moser-Festschrift, Academic Press, 1990.
[To1] Toda, M.: On the existence of H-surfaces into Riemannian manifolds. Calc. Var. 5 (1997), 55–83.
[To2] Toda, M.: Existence and non-existence results of H-surfaces into 3-dimensional Riemannian manifolds. Comm. in Analysis and Geometry 4 (1996), 161–178.
[Tom1] Tomi, F.: Ein einfacher Beweis eines Regularitätssatzes für schwache Lösungen gewisser elliptischer Systeme. Math. Z. 112 (1969), 214–218.
[Tom2] Tomi, F.: Bemerkungen zum Regularitätsproblem der Gleichung vorgeschriebener mittlerer Krümmung. Math. Z. 132 (1973), 323–326.
[Wa] Wang, G.: The Dirichlet problem for the equation of prescribed mean curvature. Ann. Inst. Henri Poincaré (Anal. Non Linéaire) 9 (1992), 643–655.
[Wen1] Wente, H.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26 (1969), 318–344.
[Wen2] Wente, H.: A general existence theorem for surfaces of constant mean curvature. Math. Z. 120 (1971), 277–288.
[Wen3] Wente, H.: An existence theorem for surfaces in equilibrium satisfying a volume constraint. Arch. Ration. Mech. Anal. 50 (1973), 139–158.
[Wer] Werner, H.: Das Problem von Douglas für Flächen konstanter mittlerer Krümmung. Math. Ann. 133 (1957), 303–319.
[Ya] Yau, S.T.: Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. Éc. Norm. Sup. 83 (1975), 487–507.
[Zi] Ziemer, W.P.: Weakly differentiable functions. Springer-Verlag, New York Berlin Heidelberg, 1989.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1999 Springer-Verlag
About this chapter
Cite this chapter
Steffen, K. (1999). Parametric surfaces of prescribed mean curvature. In: Hildebrandt, S., Struwe, M. (eds) Calculus of Variations and Geometric Evolution Problems. Lecture Notes in Mathematics, vol 1713. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092671
Download citation
DOI: https://doi.org/10.1007/BFb0092671
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65977-8
Online ISBN: 978-3-540-48813-2
eBook Packages: Springer Book Archive