Hypersurfaces of constant curvature in hyperbolic space
Abstract
We continue the work done in [2],[3] which investigates the problem of finding Weingarten hypersurfaces of constant curvature satisfying (1), (2) below in hyperbolic space $ H^{n+1}$ with a prescribed asymptotic boundary at infinity.
In [2], the focus is on the case of finding complete hypersurfaces with positive hyperbolic principal curvatures everywhere; in [3], the focus is on finding graphs over a domain with nonnegative mean curvature.
In [2] and [3], some restriction is imposed on $\sigma$ to assure us of the existence. The main aim of this article is to remove these restrictions.
The results stated in the manuscript, as well as more general ones
have been proved in [4] and [5] with a less elementary approach.
Keywords
Full Text:
PDFReferences
Cafferelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations III:functions of eigenvalues of the Hessians. Acta Math. {bf 155}, 261-301 (1985).
Guan, B., Spruck, J., Szapiel, M: Hypersurfaces of constant curvature in hyperbolic space I. J. Geom. Anal. 19 (2009), 772-795.
Guan, B., Spruck, J: Hypersurfaces of constant curvature in hyperbolic space II. J. Eur. Math. Soc. 12 (2010), 797-817.
Guan, B.; Spruck, J., Convex hypersurfaces of constant curvature in hyperbolic space. Surveys in geometric analysis and relativity, 241–257,
Adv. Lect. Math. (ALM), 20, Int. Press, Somerville, MA, 2011.
Guan, B.; Spruck, J.; Xiao, L., Interior curvature estimates and the asymptotic plateau problem in hyperbolic space. J. Differential Geom. 96 (2014), no. 2, 201–222.
Refbacks
- There are currently no refbacks.
Copyright (c) 2018 Journal of Progressive Research in Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright © 2016 Journal of Progressive Research in Mathematics. All rights reserved.
ISSN: 2395-0218.
For any help/support contact us at editorial@scitecresearch.com, jprmeditor@scitecresearch.com.