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
[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