Hyperbolic Ricci-Bourguignon-Harmonic Flow

Document Type : Original Scientific Paper

Author

Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

Abstract

In this paper, we consider hyperbolic Ricci-Bourguignon flow on a compact Riemannian manifold M coupled with the harmonic map flow between M and a fixed manifold N. At the first, we prove the unique short-time existence to solution of this system. Then, we find the second variational of some geometric structure of M along this system such as, curvature tensors. In addition, for emphasize the importance of hyperbolic Ricci-Bourguignon flow, we give some examples of this flow on Riemannian manifolds.

Keywords


[1] S. Azami, Ricci-Bourguignon flow coupled with harmonic map, Int. J. Math. 30 (10) (2019) 1950049.
[2] S. Azami, Harmonic-hyperbolic geometric flow, Electron. J. Diff. Equ. 2017 (165) (2017) 1 − 9.
[3] S. Brendle and R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc. 22 (1) (2009) 287 − 307.
[4] G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza and L. Mazzieri, The Ricci-Bourguignon flow, Pacific J. Math. 287 (2) (2015) 337 − 370.
[5] W. R. Dai, D. X. Kong and K. Liu, Hyperbolic geometric flow (I): short-time existence and nonlinear stability, Pure Appl. Math. Q. 6 (2010) 331 − 359.
[6] D. DeTurck, Deforming metrics in direction of their Ricci tensors, J. Diff. Geom. 18 (1983) 157 − 162.
[7] J. Eells and J. Sampson, Harmonic mapping of Riemannian manifolds, Amer. J. Math. 86 (1964) 109 − 169.
[8] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255 − 306.
[9] S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980) 43 − 101.
[10] B. List, Evolution of an extended Ricci flow system, Comm. Anal Geom. 16 (5) (2008) 1007 − 1048.
[11] R. Müller, Ricci flow coupled with harmonic map flow, Ann. Sci. Éc. Norm. Supér 45 (2012) 101 − 142.
[12] J. Nash, The embedding problem for Riemannian manifolds, Ann. Math. 63 (2) (1956) 20 − 63.
[13] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, Arxiv:math/0211159v1 (2002).