Generalized Ricci flow I: Local existence and uniqueness. (English) Zbl 1182.35145
Lin, Kevin (ed.) et al., Topology and physics. Proceedings of the Nankai international conference in memory of Xiao-Song Lin, Tianjin, China, July 27–31, 2007. Hackensack, NJ: World Scientific (ISBN 978-981-281-910-9/hbk). Nankai Tracts in Mathematics 12, 151-171 (2008).
Summary: We investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential equations are strictly and uniformly parabolic. Based on this, we show that the generalized Ricci flow defined on an \(n\)-dimensional compact Riemannian manifold admits a unique short-time smooth solution. Moreover, we also derive the evolution equations for the curvatures, which play an important role in our future study.
For the entire collection see [Zbl 1154.82001].
For the entire collection see [Zbl 1154.82001].
MSC:
| 35K59 | Quasilinear parabolic equations |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 57M40 | Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
