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  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.

Citation: Qiang Tu. A class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $. Discrete & Continuous Dynamical Systems, 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.  Google Scholar

[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.   Google Scholar

[3]

L. CaffarelliL. 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.  Google Scholar

[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.  Google Scholar

[5]

F. J. de AndradeJ. 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.  Google Scholar

[6]

J. M. EspinarJ. 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.  Google Scholar

[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.  Google Scholar

[8]

W. J. Firey, Christoffel's problem for general convex bodies, Mathematika, 15 (1968), 7-21.  doi: 10.1112/S0025579300002321.  Google Scholar

[9]

B. Guan and P. Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math., 156 (2002), 655-673.  doi: 10.2307/3597202.  Google Scholar

[10]

P. GuanJ. Li and Y. Li, Hypersurfaces of prescribed curvature measure, Duke Math. J., 161 (2012), 1927-1942.  doi: 10.1215/00127094-1645550.  Google Scholar

[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.  Google Scholar

[12]

P. GuanC. 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.  Google Scholar

[13]

Y. Hu, H. Li and Y. Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, preprint, arXiv: 2002.10643. Google Scholar

[14]

N. M. Ivochkina, The Dirichlet problem for the equations of curvature of order $m$, Leningrad Math. J., 2 (1991), 631-654.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. Ren and Z. Wang, The global curvature estimates for the $n-2$ Hessian equation, preprint, arXiv: 2002.08702. Google Scholar

[22]

C. Ren and Z. Wang, Notes on the curvature estimates for Hessian equations, preprint, arXiv: 2003.14234. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[3]

L. CaffarelliL. 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.  Google Scholar

[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.  Google Scholar

[5]

F. J. de AndradeJ. 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.  Google Scholar

[6]

J. M. EspinarJ. 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.  Google Scholar

[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.  Google Scholar

[8]

W. J. Firey, Christoffel's problem for general convex bodies, Mathematika, 15 (1968), 7-21.  doi: 10.1112/S0025579300002321.  Google Scholar

[9]

B. Guan and P. Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math., 156 (2002), 655-673.  doi: 10.2307/3597202.  Google Scholar

[10]

P. GuanJ. Li and Y. Li, Hypersurfaces of prescribed curvature measure, Duke Math. J., 161 (2012), 1927-1942.  doi: 10.1215/00127094-1645550.  Google Scholar

[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.  Google Scholar

[12]

P. GuanC. 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.  Google Scholar

[13]

Y. Hu, H. Li and Y. Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, preprint, arXiv: 2002.10643. Google Scholar

[14]

N. M. Ivochkina, The Dirichlet problem for the equations of curvature of order $m$, Leningrad Math. J., 2 (1991), 631-654.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. Ren and Z. Wang, The global curvature estimates for the $n-2$ Hessian equation, preprint, arXiv: 2002.08702. Google Scholar

[22]

C. Ren and Z. Wang, Notes on the curvature estimates for Hessian equations, preprint, arXiv: 2003.14234. Google Scholar

[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.  Google Scholar

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