Hostname: page-component-77f85d65b8-2tv5m Total loading time: 0 Render date: 2026-03-28T13:06:47.353Z Has data issue: false hasContentIssue false
  • English
  • Français

A class of Hessian quotient equations in de Sitter space

Part of: Global differential geometry Elliptic equations and systems

Published online by Cambridge University Press:  06 March 2024

Jinyu Gao ,
Guanghan Li  and
Kuicheng Ma
Jinyu Gao
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China e-mail: jinyugao@whu.edu.cn ghli@whu.edu.cn
Guanghan Li
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China e-mail: jinyugao@whu.edu.cn ghli@whu.edu.cn
Kuicheng Ma*
Affiliation:
School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, China
*
e-mail: kchm@sdut.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider the closed spacelike solution to a class of Hessian quotient equations in de Sitter space. Under mild assumptions, we obtain an existence result using standard degree theory based on a priori estimates.

Keywords

Hessian quotientde Sitter spacedegree theorycurvature estimate

MSC classification

Primary: 35J60: Nonlinear elliptic equations 53C42: Immersions (minimal, prescribed curvature, tight, etc.)

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 3 , September 2024 , pp. 805 - 821
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Andrade, F., Barbosa, J., and de Lira, J., Closed Weingarten hypersurfaces in warped product manifolds . Indiana Univ. Math. J. 58(2009), no. 4, 1691–1718.CrossRefGoogle Scholar
Ballesteros-Chávez, D., Klingenberg, W., and Lambert, B., Prescribed $k$ -symmetric curvature hypersurfaces in de Sitter space . Canad. Math. Bull 64(2021), 886–901.CrossRefGoogle Scholar
Caffarelli, L., Nirenberg, L., and Spruck, J., Nonlinear second order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces . In: Y. Ohya, K. Kasahara and N. Shimakura (eds.), Current topics in partial differential equations, Kinokunize Co., Tokyo, 1986, pp. 1–26.Google Scholar
Chen, D., Li, H., and Wang, Z., Starshaped compact hypersurfaces with prescribed Weingarten curvature in warped product manifolds . Calc. Var. Partial Differential Equations 57(2018), no. 2, Article no. 42.CrossRefGoogle Scholar
Chen, L., Tu, Q., and Xiang, N., Pogorelov type estimates for a class of Hessian quotient equations . J. Differential Equations 282(2021), 272–284.CrossRefGoogle Scholar
Chen, X., Tu, Q., and Xiang, N., A class of Hessian quotient equations in Euclidean space . J. Differential Equations 269(2020), no. 12, 11172–11194.CrossRefGoogle Scholar
Chen, X., Tu, Q., and Xiang, N., A class of Hessian quotient equations in the warped product manifold, 2021. arXiv:2105.12047v1 Google Scholar
Chu, J. and Jiao, H., Curvature estimates for a class of Hessian type equations . Calc. Var. Partial Differential Equations 60(2021), no. 3, 90.CrossRefGoogle Scholar
Ecker, K. and Huisken, G., Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes . Comm. Math. Phys. 135(1991), no. 3, 595–613.CrossRefGoogle Scholar
Evans, L., Classical solutions of fully nonlinear, convex, second-order elliptic equations . Comm. Pure Appl. Math. 35(1982), no. 3, 333–363.CrossRefGoogle Scholar
Gerhardt, C., Closed hypersurfaces of prescribed mean curvature in locally conformally flat Riemannian manifolds . J. Differential Geom. 48(1998), no. 3, 587–613.CrossRefGoogle Scholar
Gerhardt, C., Hypersurfaces of prescribed curvature in Lorentzian manifolds . Indiana Univ. Math. J. 49(2000), 1125–1153.CrossRefGoogle Scholar
Guan, P., Li, J., and Li, Y., Hypersurfaces of prescribed curvature measure . Duke Math. J. 161(2012), no. 10, 1927–1942.CrossRefGoogle Scholar
Guan, P., Ren, C., and Wang, Z., Global ${C}^2$ estimates for convex solutions of curvature equations . Comm. Pure Appl. Math. 68(2015), no. 8, 1287–1325.CrossRefGoogle Scholar
Jin, Q. and Li, Y., Starshaped compact hypersurfaces with prescribed $k$ -th mean curvature in hyperbolic space . Discrete Contin. Dyn. Syst. 15(2006), 367–377.CrossRefGoogle Scholar
Krylov, N., Boundedly inhomogeneous elliptic and parabolic equations in a domain . Izv. Akad. Nauk SSSR Ser. Mat. 47(1983), no. 1, 75–108.Google Scholar
Li, Y., Degree theory for second order nonlinear elliptic operators and its applications . Comm. Partial Differential Equations 14(1989), no. 11, 1541–1578.Google Scholar
Oliker, V., Hypersurfaces in ${\mathbb{R}}^{n+1}$ with prescribed Gaussian curvature and related equations of Monge–Ampére type . Comm. Partial Differential Equations 9(1984), no. 8, 807–838.CrossRefGoogle Scholar
Ren, C. and Wang, Z., On the curvature estimates for Hessian equations . Amer. J. Math. 141(2019), no. 5, 1281–1315.CrossRefGoogle Scholar
Spruck, J. and Xiao, L., A note on starshaped compact hypersurfaces with prescribed scalar curvature in space form . Rev. Mat. Iberoam. 33(2017), no. 2, 547–554.CrossRefGoogle Scholar
Yang, F., Prescribed curvature measure problem in hyperbolic space . Comm. Pure Appl. Math. 77(2024), no. 1, 863–898.CrossRefGoogle Scholar