Abstract
By means of a weight matrix, we introduce the class of weighted minimal hypersurfaces which yield a natural generalisation of minimal surfaces. Generalising a classical result of Radó, we prove existence, uniqueness and graph representation for weighted minimal hypersurfaces in \({\mathbb{R}^{n+1}}\) with prescribed boundary manifold.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bergner M., Dittrich J.: On surfaces of prescribed weighted mean curvature. Calc. Var. 33, 169–185 (2008)
Clarenz U., von der Mosel H.: Compactness theorems and an isoperimetric inequality for critical points of elliptic parametric functionals. Calc. Var. 12, 85–107 (2001)
Clarenz U.: Enclosure theorems for extremals of elliptic parametric functionals. Calc. Var. 15, 313–324 (2002)
Clarenz, U., von der Mosel, H.: On surfaces of prescribed F-mean curvature. Pacific J. Math. 213(1) (2004)
Dierkes, U., Hildebrandt, S., Küüster, W., Wohlrab, 0.: Minimal Surfaces I, II. Springer, Berlin, Heidelberg, New York (1992)
Föhlich S.: On two-dimensional immersions that are stable for parametric functionals of constant mean curvature type. Differ. Geom. Appl. 23, 235–256 (2005)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Heidelberg, New York (1983)
Hildebrandt S.: Maximum principles for minimal surfaces and for surfaces of continuous mean curvature. Math. Z. 128, 253–269 (1972)
Hildebrandt S., von der Mosel H.: Plateau’s problem for parametric double integrals. Part I: existence and regularity in the interior. Comm. Pure Appl. Math. 56, 926–955 (2003)
Hildebrandt S., von der Mosel H.: Plateau’s problem for parametric double integrals. Part II: regularity at the boundary. J. Reine Angew. Math. 565, 207–233 (2003)
Kneser H.: Lösung der Aufgabe 41. Jahresber. DMV 35, 123–124 (1926)
Nitsche, J.C.C.: Vorlesungen über Minimalflächen. Die Grundlehren der mathematischen Wissenschaften, vol. 199, 775 pp. Springer, Berlin (1975)
Rado T.: The problem of the least area and the problem of Plateau. Math. Z. 32, 763–796 (1930)
Sauvigny F.: Flächen vorgeschriebener mittlerer Krümmung mit eineindeutiger Projektion auf eine Ebene. Math. Z. 180, 41–67 (1982)
Sauvigny F.: Curvature estimates for immersions of minimal surface type via uniformizatin and theorems of Bernstein type. Manuscripta Math. 67, 567–582 (1990)
Sauvigny F.: Deformation of boundary value problems for surfaces of prescribed mean curvature. Analysis 21, 157–169 (2000)
Sauvigny, F.: Partial Differential Equations, Vols. 1 and 2. Springer Universitext, Berlin (2006)
White B.: Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on 3-manifolds. J. Differ. Geom. 33, 413–443 (1990)
Winklmann S.: Existence and Uniqueness of F-Minimal Surfaces. Ann. Glob. Anal. Geom. 24, 269–277 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bergner, M., Fröhlich, S. Existence, uniqueness and graph representation of weighted minimal hypersurfaces. Ann Glob Anal Geom 36, 363–373 (2009). https://doi.org/10.1007/s10455-009-9164-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-009-9164-x