A class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in <inline-formula><tex-math id="M1">\begin{document} \mathbb{H}^{n+1} \end{document}</tex-math></inline-formula>

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November 2021, 41(11): 5397-5407. doi: 10.3934/dcds.2021081

A class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $

Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China

^* Corresponding author: Qiang Tu

Received September 2020 Revised March 2021 Published November 2021 Early access May 2021

Fund Project: The first author is supported by Natural Science Foundation of Hubei Province, China, No. 2020CFB246

Inspired by the generalized Christoffel problem, we consider a class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $. Under some sufficient conditions, we prove the a priori estimates for solutions to the Monge-Ampère type equation $ \det(\kappa-\mathbf{1}) = f(X, \nu(X)) $. Moreover, we obtain an existence result for the compact horo-convex hypersurface $ M $ satisfying the above equation with various assumptions.

Keywords: Shifted Gauss curvature, horo-convex, Mong-Ampère type equation.

Mathematics Subject Classification: Primary: 35J96, 53C45; Secondary: 53A05.

Citation: Qiang Tu. A class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5397-5407. doi: 10.3934/dcds.2021081

References:

[1]	S. B. Alexander and R. J. Currier, Nonnegatively curved hypersurfaces of hyperbolic space and subharmonic functions, J. London Math. Soc., 41 (1990), 347-360. doi: 10.1112/jlms/s2-41.2.347.
[2]	I. Ja. Bakel'man and B. E. Kantor, Existence of a hypersurface homeomorphic to the sphere in Euclidean space with a given mean curvature, Geometry and Topology, 1 (1974), 3-10.
[3]	L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306.
[4]	L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces, Current Topics in Partial Differential Equations, (1986), 1–26.
[5]	F. J. de Andrade, J. L. M. Barbosa and J. H. S. de Lira, Closed Weingarten hypersurfaces in warped product manifolds, Indiana Univ. Math. J., 58 (2009), 1691-1718. doi: 10.1512/iumj.2009.58.3631.
[6]	J. M. Espinar, J. A. Gálvez and P. Mira, Hypersurfaces in $\mathbb{H}^{n+1}$ and conformally invariant equations: The generalized Christoffel and Nirenberg problems, J. Eur. Math. Soc., 11 (2009), 903-939. doi: 10.4171/JEMS/170.
[7]	L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. doi: 10.1002/cpa.3160350303.
[8]	W. J. Firey, Christoffel's problem for general convex bodies, Mathematika, 15 (1968), 7-21. doi: 10.1112/S0025579300002321.
[9]	B. Guan and P. Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math., 156 (2002), 655-673. doi: 10.2307/3597202.
[10]	P. Guan, J. Li and Y. Li, Hypersurfaces of prescribed curvature measure, Duke Math. J., 161 (2012), 1927-1942. doi: 10.1215/00127094-1645550.
[11]	P. Guan, C. Lin and X.-N. Ma, The existence of convex body with prescribed curvature measures, Int. Math. Res. Not., (2009), 1947–1975. doi: 10.1093/imrn/rnp007.
[12]	P. Guan, C. Ren and Z. Wang, Global $C^2$-estimates for convex solutions of curvature equations, Comm. Pure Appl. Math., 68 (2015), 1287-1325. doi: 10.1002/cpa.21528.
[13]	Y. Hu, H. Li and Y. Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, preprint, arXiv: 2002.10643.
[14]	N. M. Ivochkina, The Dirichlet problem for the equations of curvature of order $m$, Leningrad Math. J., 2 (1991), 631-654.
[15]	Q. Jin and Y. Li, Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space, Discrete Contin. Dyn. Syst., 15 (2006), 367-377. doi: 10.3934/dcds.2006.15.367.
[16]	Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations, 14 (1989), 1541-1578. doi: 10.1080/03605308908820666.
[17]	Y. Li and V. I. Oliker, Starshaped compact hypersurfaces with prescribed $m$-th mean curvature in elliptic space, J. Partial Differential Equations, 15 (2002), 68-80.
[18]	V. I. Oliker, Hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed Gaussian curvature and related equations of Monge-Ampére type, Comm. Partial Differential Equations, 9 (1984), 807-838. doi: 10.1080/03605308408820348.
[19]	V. I. Oliker, The Gauss curvature and Minkowski problems in space forms, Recent Developments in Geometry (Los Angeles, CA, 1987), Contemp. Math., Amer. Math. Soc., Providence, RI, 101 (1989), 107–123. doi: 10.1090/conm/101/1034975.
[20]	C. Ren and Z. Wang, On the curvature estimates for Hessian equations, Amer. J. Math., 141 (2019), 1281-1315. doi: 10.1353/ajm.2019.0033.
[21]	C. Ren and Z. Wang, The global curvature estimates for the $n-2$ Hessian equation, preprint, arXiv: 2002.08702.
[22]	C. Ren and Z. Wang, Notes on the curvature estimates for Hessian equations, preprint, arXiv: 2003.14234.
[23]	J. Spruck and L. Xiao, A note on starshaped compact hypersurfaces with prescribed scalar curvature in space form, Rev. Mat. Iberoam., 33 (2017), 547-554. doi: 10.4171/RMI/948.

show all references

References:

[1]	S. B. Alexander and R. J. Currier, Nonnegatively curved hypersurfaces of hyperbolic space and subharmonic functions, J. London Math. Soc., 41 (1990), 347-360. doi: 10.1112/jlms/s2-41.2.347.
[2]	I. Ja. Bakel'man and B. E. Kantor, Existence of a hypersurface homeomorphic to the sphere in Euclidean space with a given mean curvature, Geometry and Topology, 1 (1974), 3-10.
[3]	L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306.
[4]	L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces, Current Topics in Partial Differential Equations, (1986), 1–26.
[5]	F. J. de Andrade, J. L. M. Barbosa and J. H. S. de Lira, Closed Weingarten hypersurfaces in warped product manifolds, Indiana Univ. Math. J., 58 (2009), 1691-1718. doi: 10.1512/iumj.2009.58.3631.
[6]	J. M. Espinar, J. A. Gálvez and P. Mira, Hypersurfaces in $\mathbb{H}^{n+1}$ and conformally invariant equations: The generalized Christoffel and Nirenberg problems, J. Eur. Math. Soc., 11 (2009), 903-939. doi: 10.4171/JEMS/170.
[7]	L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. doi: 10.1002/cpa.3160350303.
[8]	W. J. Firey, Christoffel's problem for general convex bodies, Mathematika, 15 (1968), 7-21. doi: 10.1112/S0025579300002321.
[9]	B. Guan and P. Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math., 156 (2002), 655-673. doi: 10.2307/3597202.
[10]	P. Guan, J. Li and Y. Li, Hypersurfaces of prescribed curvature measure, Duke Math. J., 161 (2012), 1927-1942. doi: 10.1215/00127094-1645550.
[11]	P. Guan, C. Lin and X.-N. Ma, The existence of convex body with prescribed curvature measures, Int. Math. Res. Not., (2009), 1947–1975. doi: 10.1093/imrn/rnp007.
[12]	P. Guan, C. Ren and Z. Wang, Global $C^2$-estimates for convex solutions of curvature equations, Comm. Pure Appl. Math., 68 (2015), 1287-1325. doi: 10.1002/cpa.21528.
[13]	Y. Hu, H. Li and Y. Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, preprint, arXiv: 2002.10643.
[14]	N. M. Ivochkina, The Dirichlet problem for the equations of curvature of order $m$, Leningrad Math. J., 2 (1991), 631-654.
[15]	Q. Jin and Y. Li, Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space, Discrete Contin. Dyn. Syst., 15 (2006), 367-377. doi: 10.3934/dcds.2006.15.367.
[16]	Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations, 14 (1989), 1541-1578. doi: 10.1080/03605308908820666.
[17]	Y. Li and V. I. Oliker, Starshaped compact hypersurfaces with prescribed $m$-th mean curvature in elliptic space, J. Partial Differential Equations, 15 (2002), 68-80.
[18]	V. I. Oliker, Hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed Gaussian curvature and related equations of Monge-Ampére type, Comm. Partial Differential Equations, 9 (1984), 807-838. doi: 10.1080/03605308408820348.
[19]	V. I. Oliker, The Gauss curvature and Minkowski problems in space forms, Recent Developments in Geometry (Los Angeles, CA, 1987), Contemp. Math., Amer. Math. Soc., Providence, RI, 101 (1989), 107–123. doi: 10.1090/conm/101/1034975.
[20]	C. Ren and Z. Wang, On the curvature estimates for Hessian equations, Amer. J. Math., 141 (2019), 1281-1315. doi: 10.1353/ajm.2019.0033.
[21]	C. Ren and Z. Wang, The global curvature estimates for the $n-2$ Hessian equation, preprint, arXiv: 2002.08702.
[22]	C. Ren and Z. Wang, Notes on the curvature estimates for Hessian equations, preprint, arXiv: 2003.14234.
[23]	J. Spruck and L. Xiao, A note on starshaped compact hypersurfaces with prescribed scalar curvature in space form, Rev. Mat. Iberoam., 33 (2017), 547-554. doi: 10.4171/RMI/948.

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