Abstract
In this paper the authors discuss the existence and convexity of hypersurfaces with prescribed Weingarten curvature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aeppli, A., On the uniqueness of compact solutions for certain elliptic differential equations, Proc. Amer. Math. Soc., 11, 1960, 832–836.
Alexandrov, A. D., Uniqueness theorems for surfaces in the large I, Vestnik Leningrad Univ., 11, 1956, 5–17 = Amer. Soc. Trans. Ser. 2, 21, 1962, 341–354.
Bakelman, I. and Kantor, B., Existence of spherically homeomorphic hypersurfaces in Euclidean space with prescribed mean curvature, Geom. Topol., Leningrad, 1, 1974, 3–10.
Caffarelli, L. A. and Friedman, A., Convexity of solutions of some semilinear elliptic equations, Duke Math. J., 52, 1985, 431–455.
Caffarelli, L. A., Nirenberg, L. and Spruck, J., Nonlinear second order elliptic equations IV: Starshaped compactWeingarten hypersurfaces, Current Topics in Partial Differential Equations, Y. Ohya, K. Kasahara and N. Shimakura (eds.), Kinokunize, Tokyo, 1985, 1–26.
Chou (Tso), K. S., On the existence of convex hypersurfaces with prescribed mean curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 16, 1989, 225–243.
Delanoë, P., Plongements radiaux S n → ℝn+1 a courbure de Gauss positive prescrite, Ann. Sci. Ecole Norm. Sup. (4), 18, 1985, 635–649.
Gerhardt, C., Closed Weingarten hypersurfaces in space forms, Geometric Analysis and the Calculus of Variation, F. Fort (ed.), International Press, Boston, 1996, 71–98.
Guan, B. and Guan, P., Convex Hypersurfaces of Prescribed Curvature, Ann. of Math., 156, 2002, 655–674.
Guan, P. and Lin, C. S., On the equation det(u ij +δ ij u) = u p f(x) on S n, NCTS in Tsing-Hua University, 2000, preprint.
Guan, P. and Ma, X. N., Christoffel-Minkowski Problem I: Convexity of Solutions of a Hessian Equations, Invent. Math., 151, 2003, 553–577.
Korevaar, N. J. and Lewis, J., Convex solutions of certain elliptic equations have constant rank hessians, Arch. Ration. Mech. Anal., 91, 1987, 19–32.
Oliker, V. I., Hypersurfaces in ℝn+1 with prescribed Gaussian curvature and related equations of Monge-Ampère type, Comm. Partial Differential Equations, 9, 1984, 807–838.
Singer, I., Wong, B., Yau, S.-T. and Yau, Stephen S.-T., An estimate of gap of the first two eigenvalues in the Schrodinger operator, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 12, 1985, 319–333.
Treibergs, A., Existence and convexity of hypersurfaces of prescribed mean curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 12, 1985, 225–241.
Treibergs, A. and Wei, S. W., Embedded hypersurfaces with prescribed mean curvature, J. Differential Geom., 18, 1983, 513–521.
Yau, S.-T., Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., 102, Princeton Univ. Press, 1982, 669–706.
Zhu, X. P., Multiple convex hypersurfaces with prescribed mean curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 21, 1994, 175–191.
Author information
Authors and Affiliations
Corresponding author
Additional information
* Project supported by the NSERC Discovery Grant, the Fok Ying Tung Eduction Foundation, the National Natural Science Foundation of China (No.10371041) and Hundred Talents Program of Chinese Academy of Sciences.
Rights and permissions
About this article
Cite this article
Guan, P., Lin, C. & Ma, X. The Christoffel-Minkowski Problem II: Weingarten Curvature Equations*. Chin. Ann. Math. Ser. B 27, 595–614 (2006). https://doi.org/10.1007/s11401-005-0575-0
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11401-005-0575-0