Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-05T16:07:44.868Z Has data issue: false hasContentIssue false
  • English
  • Français

The Weyl Problem With Nonnegative Gauss Curvature In Hyperbolic Space

Published online by Cambridge University Press:  20 November 2018

Jui-En Chang  and
Ling Xiao
Jui-En Chang
Affiliation:
Department of Mathematics, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2686, USA. e-mail: jchang@math.jhu.edu
Ling Xiao
Affiliation:
Department of Mathematics, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2686, USA. e-mail: lxiao@math.jhu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$ and extend the result to $n$-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.

Keywords

53A9935J1558J05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 1 , 01 February 2015 , pp. 107 - 131
Copyright
Copyright © Canadian Mathematical Society 2015

References

[AP50] Aleksandrov, A. D. and Pogorelov, A. V., Uniqueness of convex surfaces of revolution. Mat. Sb. (N.S.) 26(68)(1950), 183–204.Google Scholar
[CNS84] Caffarell, L., Nirenberg, L., and Spruck, J., Monge–Ampere equations. Comm. Pure Appl. Math. 37(1984), 369–402.http://dx.doi.org/10.1002/cpa.3160370306 Google Scholar
[Ca58] Calabi, E., Improper affine hyperspheres of convex type and a generalization of a theorem ofK. Jordens. Michigan Math. J. 5(1958), 105–126.http://dx.doi.org/10.1307/mmj/1028998055 Google Scholar
[GL94] Guan, P. and Li, Y., The Weyl problem with nonnegative Gauss curvature. J. Differential Geom. 39(1994), 331–342.Google Scholar
[GS11] Guan, B. and Spruck, J., Convex hypersurfaces of constant curvature in hyperbolic space. In: Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM) 20, Int. Press, Somerville, MA, 2011, 241–257.Google Scholar
[GSS09] Guan, B., Spruck, J., and Szapiel, M., Hypersurfaces of constant curvature in hyperbolic space I. J. Geom. Anal. 19(2009), 772–795. http://dx.doi.org/10.1007/s12220-009-9086-7 Google Scholar
[HZ95] Hong, J. and Zuily, C., Isometric embedding of the 2-sphere with nonegative curvature in R3. Math. Z. 219(1995), 323–334. http://dx.doi.org/10.1007/BF02572368 Google Scholar
[Ia92] Iaia, J. A., Isometric embeddings of surfaces with nonnegative curvature in R3. Duke Math. J. 67(1992), 423–459. http://dx.doi.org/10.1215/S0012-7094-92-06717-2 Google Scholar
[Le35] Lewy, H., A priori limitations for solutions of Monge–Ampere equations, I. Trans. Amer. Math. Soc. 37(1935), 417–434.Google Scholar
[LW99] Li, Y. and Weinstein, G., A priori bounds for co-dimension one isometric embeddings. Amer. J. Math. 121(1999), 945–965.http://dx.doi.org/10.1353/ajm.1999.0035 Google Scholar
[Ni53] Nirenberg, Louis, The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6(1953), 337–394. http://dx.doi.org/10.1002/cpa.3160060303 Google Scholar
[Po64] Pogorelov, A. V., Some results on surface theory in the large. Advances in Math. 1(1964), 191–264.http://dx.doi.org/10.1016/0001-8708(65)90039-3 Google Scholar