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Prescribed mean curvature equation on the unit ball in the presence of reflection or rotation symmetry

Published online by Cambridge University Press:  27 December 2018

Pak Tung Ho
Pak Tung Ho*
Affiliation:
Department of Mathematics, Sogang University, Seoul 121-742, Korea ([email protected]; [email protected])
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Abstract

Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is f, when f possesses certain reflection or rotation symmetry.

Keywords

mean curvatureunit ballNirenberg's problemPrimary 53C4453A30Secondary 35J9335B44
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 781 - 794
Copyright
Copyright © Royal Society of Edinburgh 2018 

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Footnotes

Current address: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544 USA Email address: [email protected]

References

1Abdelhedi, W. and Chtioui, H.. The prescribed boundary mean curvature problem on the standard n-dimensional ball. Nonlinear Anal. 67 (2007), 668–686.Google Scholar
2Abdelhedi, W., Chtioui, H. and Ould Ahmedou, M.. Conformal metrics with prescribed boundary mean curvature on balls. Ann. Global Anal. Geom. 36 (2009), 327–362.Google Scholar
3Aubin, T. and Bismuth, S.. Courbure scalaire prescrite sur les variétés riemanniennes compactes dans le cas négatif. J. Funct. Anal. 143 (1997), 529–541.Google Scholar
4Baird, P., Fardoun, A. and Regbaoui, R.. The evolution of the scalar curvature of a surface to a prescribed function. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), 17–38.Google Scholar
5Berger, M. S.. Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds. J. Differ. Geom. 5 (1971), 325–332.Google Scholar
6Cao, D. and Peng, S.. Concentration of solutions for the Yamabe problem on half-spaces. Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 73–99.Google Scholar
7Chang, S.-Y. A. and Yang, P. C.. Prescribing Gaussian curvature on S 2. Acta Math. 159 (1987), 215–259.Google Scholar
8Chang, S.-Y. A. and Yang, P. C.. Conformal deformation of metrics on S 2. J. Differ. Geom. 27 (1988), 259–296.Google Scholar
9Chang, S.-Y. A. and Yang, P. C.. A perturbation result in prescribing scalar curvature on S n. Duke Math. J. 64 (1991), 27–69.Google Scholar
10Chang, S.-Y. A., Gursky, M. J. and Yang, P. C.. The scalar curvature equation on 2 - and 3-spheres. Calc. Var. Partial. Differ. Equ. 1 (1993), 205–229.Google Scholar
11Chang, S.-Y. A., Xu, X. and Yang, P. C.. A perturbation result for prescribing mean curvature. Math. Ann. 310 (1998), 473–496.Google Scholar
12Chen, X. and Ho, P. T.. Conformal curvature flows on compact manifold of negative Yamabe constant. Indiana U. Math. J. 67 (2018), 537–581.Google Scholar
13Chen, W. and Li, C.. Prescribing scalar curvature on S n. Pacific J. Math. 199 (2001), 61–78.Google Scholar
14Chen, W. and Li, C.. Gaussian curvature in the negative case. Proc. Amer. Math. Soc. 131 (2003), 741–744.Google Scholar
15Chen, X. and Xu, X.. The scalar curvature flow on S n—perturbation theorem revisited. Invent. Math. 187 (2012), 395–506.Google Scholar
16Chen, X., Ho, P. T. and Sun, L.. Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant. Ann. Global Anal. Geom. 53 (2018), 121–150.Google Scholar
17Cherrier, P.. Problèmes de Neumann non linéaires sur les variétés riemanniennes. J. Funct. Anal. 57 (1984), 154–207.Google Scholar
18Chtioui, H., Ould Ahmedou, M. and Yacoub, R.. Topological methods for the prescribed Webster scalar curvature problem on CR manifolds. Differ. Geom. Appl. 28 (2010), 264–281.Google Scholar
19Chtioui, H., Ould Ahmedou, M. and Yacoub, R.. Existence and multiplicity results for the prescribed Webster scalar curvature problem on three CR manifolds. J. Geom. Anal. 23 (2013), 878–894.Google Scholar
20Djadli, Z., Malchiodi, A. and Ould Ahmedou, M.. Prescribing scalar and boundary mean curvature on the three dimensional half sphere. J. Geom. Anal. 13 (2003), 255–289.Google Scholar
21Escobar, J. F.. Conformal metrics with prescribed mean curvature on the boundary. Calc. Var. Partial. Differ. Equ. 4 (1996), 559–592.Google Scholar
22Escobar, J. F. and Garcia, G.. Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary. J. Funct. Anal. 211 (2004), 71–152.Google Scholar
23Escobar, J. F. and Schoen, R.. Conformal metrics with prescribed scalar curvature. Invent. Math. 86 (1986), 243–254.Google Scholar
24Han, Z. C.. Prescribing Gaussian curvature on S 2. Duke Math. J. 61 (1990), 679–703.Google Scholar
25Han, Z. C. and Li, Y. Y.. The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature. Comm. Anal. Geom. 8 (2000), 809–869.Google Scholar
26Han, Z. C. and Li, Y. Y.. On the local solvability of the Nirenberg problem on S 2. Discrete Contin. Dyn. Syst. 28 (2010), 607–615.Google Scholar
27Ho, P. T.. Prescribed curvature flow on surfaces. Indiana Univ. Math. J. 60 (2011), 1517–1542.Google Scholar
28Ho, P. T.. Results related to prescribing pseudo-Hermitian scalar curvature. Int. J. Math. 24 (2013), 29pp.Google Scholar
29Ho, P. T.. The Webster scalar curvature flow on CR sphere. Part I. Adv. Math. 268 (2015a), 758–835.Google Scholar
30Ho, P. T.. The Webster scalar curvature flow on CR sphere. Part II. Adv. Math. 268 (2015b), 836–905.Google Scholar
31Ho, P. T.. Prescribed Webster scalar curvature on S 2n+1 in the presence of reflection or rotation symmetry. Bull. Sci. Math. 140 (2016), 506–518.Google Scholar
32Kazdan, J. L. and Warner, F. W.. Curvature functions for compact 2-manifolds. Ann. of Math. (2) 99 (1974), 14–47.Google Scholar
33Kazdan, J. L. and Warner, F. W.. Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures. Ann. of Math. (2) 101 (1975), 317–331.Google Scholar
34Leung, M. C. and Zhou, F.. Prescribed scalar curvature equation on S n in the presence of reflection or rotation symmetry. Proc. Amer. Math. Soc. 142 (2014), 1607–1619.Google Scholar
35Moser, J.. On a nonlinear problem in differential geometry. Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 273–280 (New York: Academic Press, 1973).Google Scholar
36Ngô, Q. A. and Zhang, H.. Prescribing Webster scalar curvature on CR manifolds of negative conformal invariants. J. Differ. Equ. 258 (2015), 4443–4490.Google Scholar
37Ouyang, T.. On the positive solutions of semilinear equations Δu+λu−hu p=0 on the compact manifolds. Trans. Amer. Math. Soc. 331 (1992), 503–527.Google Scholar
38Schoen, R. and Zhang, D.. Prescribed scalar curvature on the n-sphere. Calc. Var. Partial. Differ. Equ. 4 (1996), 1–25.Google Scholar
39Sharaf, K.. On the boundary mean curvature equation on 𝔹n. Bound. Value Probl. 2016 (2016), 221.Google Scholar
40Sharaf, K., Alharthy, H. and Altharwi, S.. Conformal transformation of metrics on the n-ball. Nonlinear Anal. 95 (2014), 246–262.Google Scholar
41Struwe, M.. A flow approach to Nirenberg's problem. Duke Math. J. 128 (2005), 19–64.Google Scholar
42Tang, J. J.. Solvability of the equation $\Delta_{g} u+{\tilde S}u^{\sigma}=Su$ on manifolds. Proc. Amer. Math. Soc. 121 (1994), 83–92.Google Scholar
43Wei, J. C. and Xu, X.. On conformal deformations of metrics on S n. J. Funct. Anal. 157 (1998), 292–325.Google Scholar
44Xu, X. and Yang, P. C.. Remarks on prescribing Gauss curvature. Trans. Amer. Math. Soc. 336 (1993), 831–840.Google Scholar
45Xu, X. and Zhang, H.. Conformal metrics on the unit ball with prescribed mean curvature. Math Ann. 365 (2016), 497–557.Google Scholar
46Zhang, H.. Evolution of curvatures on a surface with boundary to prescribed functions. Manuscripta Math. 149 (2016), 153–170.Google Scholar