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June  2020, 19(6): 3223-3231. doi: 10.3934/cpaa.2020140

Ricci curvature of conformal deformation on compact 2-manifolds

Yoon-Tae Jung , Soo-Young Lee and  Eun-Hee Choi ,

Department of Mathematics, Chosun University, Kwangju, 61452, Republic of Korea

* Corresponding author

Received  June 2019 Revised  December 2019 Published  March 2020

Fund Project: The first author was supported by Chosun University Research Fund 2018

In this paper, we consider Ricci curvature of conformal deformation on compact 2-manifolds. And we prove that, by the conformal deformation, the resulting manifold is an Einstein manifold.

Keywords: Conformal deformation, Ricci curvature, Einstein manifold.
Mathematics Subject Classification: Primary: 53C21, 58E30; Secondary: 53C25.58H15, 58J05.
Citation: Yoon-Tae Jung, Soo-Young Lee, Eun-Hee Choi. Ricci curvature of conformal deformation on compact 2-manifolds. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3223-3231. doi: 10.3934/cpaa.2020140
References:
[1]

T. Aubin, Nonlinear Analysis on Manifolds, Springer-Verlag, New York, 1982.  Google Scholar

[2]

M. S. Berger, Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, J. Differ. Geom., 5 (1971), 325-332.   Google Scholar

[3]

A. L. Besse, Einstein Manifolds, Springer-Verlag, New York, 1987. doi: 10.1007/978-3-540-74311-8.  Google Scholar

[4]

H. Ge and W. Jiang, Kazdan-Warner equation on infinite graph, J. Korean Math. Soc., 55 (2018), 1091-1101.  doi: 10.4134/JKMS.j170561.  Google Scholar

[5]

J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. Math., 99 (1974), 14-47.  doi: 10.2307/1971012.  Google Scholar

[6]

B. O'Neill, Semi-Riemannian Geometry, Academic, New York, 1983.  Google Scholar

[7]

R. Walter, Real and Complex Analysis, McGraw-Hill, Singapore, 1986. Google Scholar

show all references

References:
[1]

T. Aubin, Nonlinear Analysis on Manifolds, Springer-Verlag, New York, 1982.  Google Scholar

[2]

M. S. Berger, Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, J. Differ. Geom., 5 (1971), 325-332.   Google Scholar

[3]

A. L. Besse, Einstein Manifolds, Springer-Verlag, New York, 1987. doi: 10.1007/978-3-540-74311-8.  Google Scholar

[4]

H. Ge and W. Jiang, Kazdan-Warner equation on infinite graph, J. Korean Math. Soc., 55 (2018), 1091-1101.  doi: 10.4134/JKMS.j170561.  Google Scholar

[5]

J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. Math., 99 (1974), 14-47.  doi: 10.2307/1971012.  Google Scholar

[6]

B. O'Neill, Semi-Riemannian Geometry, Academic, New York, 1983.  Google Scholar

[7]

R. Walter, Real and Complex Analysis, McGraw-Hill, Singapore, 1986. Google Scholar

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