No Access

Ricci-Bourguignon flow coupled with harmonic map flow

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

https://doi.org/10.1142/S0129167X19500496Cited by:5 (Source: Crossref)

Abstract

In this paper, we study a coupled system of the Ricci–Bourguignon flow on a closed Riemannian manifold $M$ with the harmonic map flow. At the first, we will investigate the existence and uniqueness for solution of this flow on a closed Riemannian manifold and then we find evolution of some geometric structures of manifold along this flow.

Keywords:

AMSC: 53C43, 53C44, 53C21

References

1. B. Andrews and C. Hopper, The Ricci Flow in Riemannian Geometry, A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem (Springer, 2011). Crossref, Google Scholar
2. J. P. Bourguignon, Ricci Curvature and Einstein Metrics, Global Differential Geometry and Global Analysis (Berlin, 1979) Lecture Notes in Mathematics, Vol. 838 (Springer, Berlin, 1981), pp. 42–63. Crossref, Google Scholar
3. G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza and L. Mazzieri, The Ricci-Bourguignon flow, Pacific J. Math. 287(2) (2015) 337–370. Crossref, Web of Science, Google Scholar
4. Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom. 33 (1991) 749–786. Crossref, Web of Science, Google Scholar
5. D. M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom. 18(1) (1983) 157–162. Crossref, Web of Science, Google Scholar
6. J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Am. J. Math. 86 (1964) 109–169. Crossref, Web of Science, Google Scholar
7. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255–306. Crossref, Web of Science, Google Scholar
8. R. S. Hamilton, Four-manifolds with positive curvature operator, J. Diff. Geom. 24(2) (1986) 153–179. Crossref, Web of Science, Google Scholar
9. J. Jost, Riemannian Geometry and Geometric Analysis, 6th edn. (Springer, 2011). Crossref, Google Scholar
10. J. Jost, Partial Differential Equations (Springer, 2002). Google Scholar
11. R. Müller, Ricci flow coupled with harmonic map flow, Ann. Sci. Norm. Sup. 45 (2012) 101–142. Crossref, Web of Science, Google Scholar
12. R. Müller, Differential Harnack Inequalities and the Ricci Flow, Series of Lectures in Matheamatics (EMS, 2006). Crossref, Google Scholar
13. J. Nash, The embedding problem for Riemannian manifolds, Comm. Pure Appl. Math. 63(2) (1956) 20–63. Google Scholar
14. P. Topping, Lectures on the Ricci Flow, London Mathematical Society Lecture Note Series, Vol. 325 (Cambridge University Press, 2010). Google Scholar