Prescribing curvatures on three dimensional Riemannian manifolds with boundaries
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Abstract:
Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset of $\bar M$; some of them may appear on $\partial M$. We describe the asymptotic behavior of the blow-up solutions around each blow-up point and derive an energy estimate as a consequence.References
- Antonio Ambrosetti, YanYan Li, and Andrea Malchiodi, On the Yamabe problem and the scalar curvature problems under boundary conditions, Math. Ann. 322 (2002), no. 4, 667–699. MR 1905107, DOI 10.1007/s002080100267
- Henrique Araújo, Existence and compactness of minimizers of the Yamabe problem on manifolds with boundary, Comm. Anal. Geom. 12 (2004), no. 3, 487–510. MR 2128601
- Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
- Massimiliano Berti and Andrea Malchiodi, Non-compactness and multiplicity results for the Yamabe problem on $S^n$, J. Funct. Anal. 180 (2001), no. 1, 210–241. MR 1814428, DOI 10.1006/jfan.2000.3699
- Simon Brendle, A generalization of the Yamabe flow for manifolds with boundary, Asian J. Math. 6 (2002), no. 4, 625–644. MR 1958085, DOI 10.4310/AJM.2002.v6.n4.a2
- S. Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc. 24 (2008), no 4, 951-979.
- S. Brendle, F. C. Marques, Blow-up phenomena for the Yamabe PDE II, preprint 2007.
- 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
- Luis A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are $C^{1,\alpha }$, Rev. Mat. Iberoamericana 3 (1987), no. 2, 139–162. MR 990856, DOI 10.4171/RMI/47
- Luis A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 (1989), no. 1, 55–78. MR 973745, DOI 10.1002/cpa.3160420105
- Chiun-Chuan Chen and Chang-Shou Lin, Estimates of the conformal scalar curvature equation via the method of moving planes, Comm. Pure Appl. Math. 50 (1997), no. 10, 971–1017. MR 1466584, DOI 10.1002/(SICI)1097-0312(199710)50:10<971::AID-CPA2>3.0.CO;2-D
- Chiun-Chuan Chen and Chang-Shou Lin, Estimate of the conformal scalar curvature equation via the method of moving planes. II, J. Differential Geom. 49 (1998), no. 1, 115–178. MR 1642113
- Chiun-Chuan Chen and Chang-Shou Lin, Prescribing scalar curvature on $S^N$. I. A priori estimates, J. Differential Geom. 57 (2001), no. 1, 67–171. MR 1871492
- Zindine Djadli, Andrea Malchiodi, and Mohameden Ould Ahmedou, Prescribing scalar and boundary mean curvature on the three dimensional half sphere, J. Geom. Anal. 13 (2003), no. 2, 255–289. MR 1967027, DOI 10.1007/BF02930697
- Olivier Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not. 23 (2004), 1143–1191. MR 2041549, DOI 10.1155/S1073792804133278
- Olivier Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not. 23 (2004), 1143–1191. MR 2041549, DOI 10.1155/S1073792804133278
- José F. Escobar, Conformal deformation of a Riemannian metric to a constant scalar curvature metric with constant mean curvature on the boundary, Indiana Univ. Math. J. 45 (1996), no. 4, 917–943. MR 1444473, DOI 10.1512/iumj.1996.45.1344
- José F. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math. (2) 136 (1992), no. 1, 1–50. MR 1173925, DOI 10.2307/2946545
- José F. Escobar, The Yamabe problem on manifolds with boundary, J. Differential Geom. 35 (1992), no. 1, 21–84. MR 1152225
- Veronica Felli and Mohameden Ould Ahmedou, Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z. 244 (2003), no. 1, 175–210. MR 1981882, DOI 10.1007/s00209-002-0486-7
- M. Khuri, F.C. Marques, R. Schoen, A compactness theorem for the Yamabe problem, prepint, 2007.
- Zheng-Chao Han and Yanyan Li, The Yamabe problem on manifolds with boundary: existence and compactness results, Duke Math. J. 99 (1999), no. 3, 489–542. MR 1712631, DOI 10.1215/S0012-7094-99-09916-7
- Yan Yan Li, Prescribing scalar curvature on $S^n$ and related problems. I, J. Differential Equations 120 (1995), no. 2, 319–410. MR 1347349, DOI 10.1006/jdeq.1995.1115
- YanYan Li and Lei Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003), 27–87. MR 2001065, DOI 10.1007/BF02786551
- Yan Yan Li and Lei Zhang, A Harnack type inequality for the Yamabe equation in low dimensions, Calc. Var. Partial Differential Equations 20 (2004), no. 2, 133–151. MR 2057491, DOI 10.1007/s00526-003-0230-0
- Yan Yan Li and Lei Zhang, Compactness of solutions to the Yamabe problem. II, Calc. Var. Partial Differential Equations 24 (2005), no. 2, 185–237. MR 2164927, DOI 10.1007/s00526-004-0320-7
- Yanyan Li and Meijun Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), no. 2, 383–417. MR 1369398, DOI 10.1215/S0012-7094-95-08016-8
- Fernando C. Marques, Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J. 54 (2005), no. 6, 1599–1620. MR 2189679, DOI 10.1512/iumj.2005.54.2590
- Fernando Coda Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom. 71 (2005), no. 2, 315–346. MR 2197144
- Mohameden Ould Ahmedou, On the prescribed scalar and zero mean curvature on 3-dimensional manifolds with umbilic boundary, Adv. Nonlinear Stud. 6 (2006), no. 1, 13–46. MR 2196889, DOI 10.1515/ans-2006-0102
- R. Schoen, Courses at Stanford University, 1988, and New York University, 1989.
- Richard M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. MR 994021, DOI 10.1007/BFb0089180
- Kazuaki Taira, The Yamabe problem and nonlinear boundary value problems, J. Differential Equations 122 (1995), no. 2, 316–372. MR 1355895, DOI 10.1006/jdeq.1995.1151
- Steven D. Taliaferro and Lei Zhang, Asymptotic symmetries for conformal scalar curvature equations with singularity, Calc. Var. Partial Differential Equations 26 (2006), no. 4, 401–428. MR 2235880, DOI 10.1007/s00526-005-0002-0
- Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 265–274. MR 240748
- Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. MR 125546
- Lei Zhang, Refined asymptotic estimates for conformal scalar curvature equation via moving sphere method, J. Funct. Anal. 192 (2002), no. 2, 491–516. MR 1923411, DOI 10.1006/jfan.2001.3932
Additional Information
- Lei Zhang
- Affiliation: Department of Mathematics, University of Alabama at Birmingham, 1300 University Boulevard, 452 Campbell Hall, Birmingham, Alabama 35294-1170
- Email: leizhang@math.uab.edu
- Received by editor(s): September 13, 2006
- Published electronically: February 23, 2009
- Additional Notes: The author was supported by National Science Foundation Grant 0600275 (0810902)
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 3463-3481
- MSC (2000): Primary 35J60, 53B20
- DOI: https://doi.org/10.1090/S0002-9947-09-04911-3
- MathSciNet review: 2491888