Abstract
We consider two dimensional surfaces \({X : \Omega\to\mathbb R^{n+2}, \Omega\subset \mathbb C, w=u+iv\mapsto X(w)}\) with arbitrary codimension n and prove a barrier principle for strong (possibly branched) subsolutions \({X\in C^1(\Omega, \mathbb {R}^{n+2})\cap H_{2,{\rm loc}}^2(\Omega,\mathbb R^{n+2})}\) of the integral inequality
with mean curvature functions (H k ) k=1,...,n which lie locally on one side of a supporting hypersurface S. We show under suitable assumption on the 2-mean curvature of the supporting surface S that X is locally contained in S. This generalizes a corresponding result for surfaces in \({\mathbb R^3}\) , cf. (Dierkes et al., Regularity of Minimal Surfaces, §4.4, 2010).
Similar content being viewed by others
References
Dierkes U., Hildebrandt S., Sauvigny F.: Minimal Surfaces. Springer, New York (2010)
Dierkes U., Hildebrandt S., Tromba A.J.: Regularity of Minimal Surfaces. Springer, New York (2010)
Fröhlich, S.: Surfaces in Euclidean Spaces. http://page.mi.fu-berlin.de/sfroehli/Book/book.php.html (Accessed 30 May 2012)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, New York (1998)
Hartman P., Wintner A.: On the local behavior of solutions of non-parabolic partial differential equations. Am. J. Math. 75(3), 449–476 (1953)
Jorge L.P., Tomi F.: The barrier principle for minimal submanifolds of arbitrary codimension. Ann. Global Anal. Geom. 24, 261–267 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Henkemeyer, P. A Barrier Principle for Surfaces with Prescribed Mean Curvature and Arbitrary Codimension. Results. Math. 64, 67–75 (2013). https://doi.org/10.1007/s00025-012-0297-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-012-0297-z