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May  2021, 14(5): 1757-1778. doi: 10.3934/dcdss.2020453

Compactness results for linearly perturbed Yamabe problem on manifolds with boundary

Marco Ghimenti , and  Anna Maria Micheletti

Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56126 Pisa, Italy

* Corresponding author: Marco Ghimenti

Received  February 2020 Revised  June 2020 Published  November 2020

Let $ (M,g) $ a compact Riemannian $ n $-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of $ g $ there are scalar-flat metrics that have $ \partial M $ as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term $ h_{g} $ with a negative smooth function $ \alpha, $ the set of solutions of Yamabe problem is still a compact set.

Keywords: Umbilic boundary, non umbilic boundary, Yamabe problem, compactness.
Mathematics Subject Classification: 35J65, 53C21.
Citation: Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453
References:
[1]

S. de M. Almaraz, A compactness theorem for scalar-flat metrics on manifolds with boundary, Calc. Var. Partial Differential Equations, 41 (2011), 341-386.  doi: 10.1007/s00526-010-0365-8.  Google Scholar

[2]

S. de M. Almaraz, Blow-up phenomena for scalar-flat metrics on manifolds with boundary, J. Differential Equations, 251 (2011), 1813-1840.  doi: 10.1016/j.jde.2011.04.013.  Google Scholar

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S. de M. Almaraz, An existence theorem of conformal scalar-flat metrics on manifolds with boundary, Pacific J. Math., 248 (2010), 1-22.  doi: 10.2140/pjm.2010.248.1.  Google Scholar

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S. Almaraz, O. S. de Queiroz and S. Wang, A compactness theorem for scalar-flat metrics on 3-manifolds with boundary, J. Funct. Anal., 277 (2019), 2092-2116.  doi: 10.1016/j.jfa.2019.01.001.  Google Scholar

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T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics. Springer, Berlin, 1998. doi: 10.1007/978-3-662-13006-3.  Google Scholar

[6]

S. S. Chen, Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions, preprint, arXiv: 0912.1302, (2010). Google Scholar

[7]

O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., 2004 (2004), 1143-1191.  doi: 10.1155/S1073792804133278.  Google Scholar

[8]

O. Druet and E. Hebey, Blow-up examples for second order elliptic PDEs of critical sobolev growth, Trans. Amer. Math. Soc., 357 (2005), 1915-1929.  doi: 10.1090/S0002-9947-04-03681-5.  Google Scholar

[9]

J. F. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math., 136, (1992), 1–50. doi: 10.2307/2946545.  Google Scholar

[10]

V. Felli and M. Ould Ahmedou, Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z., 244 (2003), 175-210.  doi: 10.1007/s00209-002-0486-7.  Google Scholar

[11]

M. Ghimenti and A. M. Micheletti, Compactness for conformal scalar-flat metrics on umbilic boundary manifolds, Nonlinear Anal, 200 (2020), 30 pp doi: 10.1016/j.na.2020.111992.  Google Scholar

[12]

M. Ghimenti, A. M. Micheletti and A. Pistoia, Blow-up phenomena for linearly perturbed Yamabe problem on manifolds with umbilic boundary, J. Differential Equations, 267 (2019), 587-618.  doi: 10.1016/j.jde.2019.01.023.  Google Scholar

[13]

M. Ghimenti, A. M. Micheletti and A. Pistoia, Linear perturbation of the Yamabe problem on manifolds with boundary, J. Geom. Anal., 28 (2018), 1315-1340.  doi: 10.1007/s12220-017-9864-6.  Google Scholar

[14]

G. Giraud, Sur le probléme de Dirichlet généralisé, Ann. Sci. École Norm. Sup., 46 (1929), 131-145.   Google Scholar

[15]

Z.-C. Han and Y. Li, The Yamabe problem on manifolds with boundary: Existence and compactness results, Duke Math, J., 99 (1999), 489-542.  doi: 10.1215/S0012-7094-99-09916-7.  Google Scholar

[16]

S. Kim, M. Musso and J. Wei, Compactness of scalar-flat conformal metrics on low-dimensional manifolds manifolds with constant mean curvature on boundary, preprint, arXiv: 1906.01317. Google Scholar

[17]

Y. Li and M. Zhu, Yamabe type equations on three dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.  doi: 10.1142/S021919979900002X.  Google Scholar

[18]

F. C. Marques, Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J., 54 (2005), 1599-1620.  doi: 10.1512/iumj.2005.54.2590.  Google Scholar

[19]

F. C. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differ. Geom., 71 (2005), 315-346.  doi: 10.4310/jdg/1143651772.  Google Scholar

[20]

F. C. Marques, Compactness and non compactness for Yamabe-type problems, Progress in Nonlinear Differential Equation and their Applications, 86 (2015), 121-131.  doi: 10.1007/978-3-319-19902-3_9.  Google Scholar

[21]

M. Mayer and C. B. Ndiaye, Barycenter technique and the Riemann mapping problem of Cherrier-Escobar, J. Differential Geom., 107 (2017), 519-560.  doi: 10.4310/jdg/1508551224.  Google Scholar

[22]

R. Schoen and D. Zhang, Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differ. Equ., 4 (1996), 1-25.  doi: 10.1007/BF01322307.  Google Scholar

show all references

References:
[1]

S. de M. Almaraz, A compactness theorem for scalar-flat metrics on manifolds with boundary, Calc. Var. Partial Differential Equations, 41 (2011), 341-386.  doi: 10.1007/s00526-010-0365-8.  Google Scholar

[2]

S. de M. Almaraz, Blow-up phenomena for scalar-flat metrics on manifolds with boundary, J. Differential Equations, 251 (2011), 1813-1840.  doi: 10.1016/j.jde.2011.04.013.  Google Scholar

[3]

S. de M. Almaraz, An existence theorem of conformal scalar-flat metrics on manifolds with boundary, Pacific J. Math., 248 (2010), 1-22.  doi: 10.2140/pjm.2010.248.1.  Google Scholar

[4]

S. Almaraz, O. S. de Queiroz and S. Wang, A compactness theorem for scalar-flat metrics on 3-manifolds with boundary, J. Funct. Anal., 277 (2019), 2092-2116.  doi: 10.1016/j.jfa.2019.01.001.  Google Scholar

[5]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics. Springer, Berlin, 1998. doi: 10.1007/978-3-662-13006-3.  Google Scholar

[6]

S. S. Chen, Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions, preprint, arXiv: 0912.1302, (2010). Google Scholar

[7]

O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., 2004 (2004), 1143-1191.  doi: 10.1155/S1073792804133278.  Google Scholar

[8]

O. Druet and E. Hebey, Blow-up examples for second order elliptic PDEs of critical sobolev growth, Trans. Amer. Math. Soc., 357 (2005), 1915-1929.  doi: 10.1090/S0002-9947-04-03681-5.  Google Scholar

[9]

J. F. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math., 136, (1992), 1–50. doi: 10.2307/2946545.  Google Scholar

[10]

V. Felli and M. Ould Ahmedou, Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z., 244 (2003), 175-210.  doi: 10.1007/s00209-002-0486-7.  Google Scholar

[11]

M. Ghimenti and A. M. Micheletti, Compactness for conformal scalar-flat metrics on umbilic boundary manifolds, Nonlinear Anal, 200 (2020), 30 pp doi: 10.1016/j.na.2020.111992.  Google Scholar

[12]

M. Ghimenti, A. M. Micheletti and A. Pistoia, Blow-up phenomena for linearly perturbed Yamabe problem on manifolds with umbilic boundary, J. Differential Equations, 267 (2019), 587-618.  doi: 10.1016/j.jde.2019.01.023.  Google Scholar

[13]

M. Ghimenti, A. M. Micheletti and A. Pistoia, Linear perturbation of the Yamabe problem on manifolds with boundary, J. Geom. Anal., 28 (2018), 1315-1340.  doi: 10.1007/s12220-017-9864-6.  Google Scholar

[14]

G. Giraud, Sur le probléme de Dirichlet généralisé, Ann. Sci. École Norm. Sup., 46 (1929), 131-145.   Google Scholar

[15]

Z.-C. Han and Y. Li, The Yamabe problem on manifolds with boundary: Existence and compactness results, Duke Math, J., 99 (1999), 489-542.  doi: 10.1215/S0012-7094-99-09916-7.  Google Scholar

[16]

S. Kim, M. Musso and J. Wei, Compactness of scalar-flat conformal metrics on low-dimensional manifolds manifolds with constant mean curvature on boundary, preprint, arXiv: 1906.01317. Google Scholar

[17]

Y. Li and M. Zhu, Yamabe type equations on three dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.  doi: 10.1142/S021919979900002X.  Google Scholar

[18]

F. C. Marques, Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J., 54 (2005), 1599-1620.  doi: 10.1512/iumj.2005.54.2590.  Google Scholar

[19]

F. C. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differ. Geom., 71 (2005), 315-346.  doi: 10.4310/jdg/1143651772.  Google Scholar

[20]

F. C. Marques, Compactness and non compactness for Yamabe-type problems, Progress in Nonlinear Differential Equation and their Applications, 86 (2015), 121-131.  doi: 10.1007/978-3-319-19902-3_9.  Google Scholar

[21]

M. Mayer and C. B. Ndiaye, Barycenter technique and the Riemann mapping problem of Cherrier-Escobar, J. Differential Geom., 107 (2017), 519-560.  doi: 10.4310/jdg/1508551224.  Google Scholar

[22]

R. Schoen and D. Zhang, Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differ. Equ., 4 (1996), 1-25.  doi: 10.1007/BF01322307.  Google Scholar

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