An Obata-type formula and the Liouville-type theorem for a class of K-Hessian equations on the sphere
HTML articles powered by AMS MathViewer
- by Shujun Shi, Peihe Wang, Tian Wu and Hua Zhu;
- Proc. Amer. Math. Soc. 152 (2024), 3537-3550
- DOI: https://doi.org/10.1090/proc/16857
- Published electronically: June 21, 2024
- HTML | PDF | Request permission
Abstract:
In this paper, we study a class of $k$-Hessian equations, we can deduce an Obata-type formula and a Liouville-type theorem by integration by parts.References
- Marie-Françoise Bidaut-Véron and Laurent Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), no. 3, 489–539. MR 1134481, DOI 10.1007/BF01243922
- Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
- Sun-Yung A. Chang, Matthew J. Gursky, and Paul Yang, An a priori estimate for a fully nonlinear equation on four-manifolds, J. Anal. Math. 87 (2002), 151–186. Dedicated to the memory of Thomas H. Wolff. MR 1945280, DOI 10.1007/BF02868472
- Lars Gȧrding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957–965. MR 113978, DOI 10.1512/iumj.1959.8.58061
- B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. MR 615628, DOI 10.1002/cpa.3160340406
- María del Mar González, Removability of singularities for a class of fully non-linear elliptic equations, Calc. Var. Partial Differential Equations 27 (2006), no. 4, 439–466. MR 2263673, DOI 10.1007/s00526-006-0026-0
- María del Mar González, Singular sets of a class of locally conformally flat manifolds, Duke Math. J. 129 (2005), no. 3, 551–572. MR 2169873, DOI 10.1215/S0012-7094-05-12934-9
- María del Mar González, Classification of singularities for a subcritical fully nonlinear problem, Pacific J. Math. 226 (2006), no. 1, 83–102. MR 2247857, DOI 10.2140/pjm.2006.226.83
- Matthew J. Gursky and Jeff Viaclovsky, Convexity and singularities of curvature equations in conformal geometry, Int. Math. Res. Not. , posted on (2006), Art. ID 96890, 43. MR 2219225, DOI 10.1155/IMRN/2006/96890
- Zheng-Chao Han, Local pointwise estimates for solutions of the $\sigma _2$ curvature equation on 4-manifolds, Int. Math. Res. Not. 79 (2004), 4269–4292. MR 2126828, DOI 10.1155/S1073792804141743
- Zheng-Chao Han, A Kazdan-Warner type identity for the $\sigma _k$ curvature, C. R. Math. Acad. Sci. Paris 342 (2006), no. 7, 475–478 (English, with English and French summaries). MR 2214598, DOI 10.1016/j.crma.2006.01.023
- John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91. MR 888880, DOI 10.1090/S0273-0979-1987-15514-5
- Aobing Li and YanYan Li, On some conformally invariant fully nonlinear equations, C. R. Math. Acad. Sci. Paris 334 (2002), no. 4, 305–310 (English, with English and French summaries). MR 1891008, DOI 10.1016/S1631-073X(02)02264-1
- Aobing Li and Yan Yan Li, A Liouville type theorem for some conformally invariant fully nonlinear equations, Geometric analysis of PDE and several complex variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 321–328. MR 2126479, DOI 10.1090/conm/368/06788
- Yan Yan Li, Conformally invariant fully nonlinear elliptic equations and isolated singularities, J. Funct. Anal. 233 (2006), no. 2, 380–425. MR 2214582, DOI 10.1016/j.jfa.2005.08.009
- Morio Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258. MR 303464
- Qianzhong Ou, Singularities and Liouville theorems for some special conformal Hessian equations, Pacific J. Math. 266 (2013), no. 1, 117–128. MR 3105779, DOI 10.2140/pjm.2013.266.117
- Robert C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J. 20 (1973), 373–383. MR 334045
- Joel Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 283–309. MR 2167264
- Jeff A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J. 101 (2000), no. 2, 283–316. MR 1738176, DOI 10.1215/S0012-7094-00-10127-5
Bibliographic Information
- Shujun Shi
- Affiliation: School of Mathematical Sciences, Harbin Normal University, Harbin 150025, HeiLongjiang PROVINCE, People’s Republic of China
- ORCID: 0000-0002-3880-7884
- Email: shjshi@hrbnu.edu.cn
- Peihe Wang
- Affiliation: School of Mathematical Sciences, Qufu Normal University, Qufu 273165, ShanDong PROVINCE, People’s Republic of China
- Email: peihewang@hotmail.com
- Tian Wu
- Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, AnHui PROVINCE, People’s Republic of China
- Email: wt1997@mail.ustc.edu.cn
- Hua Zhu
- Affiliation: School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang 621010, SiChuan PROVINCE, People’s Republic of China; \normalfont and School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, AnHui PROVINCE, People’s Republic of China
- ORCID: 0009-0003-1002-0240
- Email: zhuhmaths@mail.ustc.edu.cn
- Received by editor(s): September 18, 2023
- Received by editor(s) in revised form: January 8, 2024, March 1, 2024, and March 2, 2024
- Published electronically: June 21, 2024
- Additional Notes: The first author was supported by the National Natural Science Foundation of China under Grant 11971137. The fourth author was supported by the National Natural Science Foundation of China under Grant 12301098.
- Communicated by: Jiaping Wang
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3537-3550
- MSC (2020): Primary 35J60; Secondary 35B45
- DOI: https://doi.org/10.1090/proc/16857
- MathSciNet review: 4767282