### Citations

45 | Gaussian densities and stability for some Ricci solitons - Cao, Hamilton, et al. |

40 | Entropy and reduced distance for Ricci expanders - Feldman, Ilmanen, et al. - 2005 |

28 | Woolgar A Gradient Flow for Worldsheet Nonlinear Sigma Models, Nucl.Phys. B739
- Oliynyk, Suneeta, et al.
- 2006
(Show Context)
Citation Context ...gical and geometric problems, people study some analogues flows, including the harmonic-Ricci flow[9, 11], connection Ricci flow[14], Ricci-Yang-Mills flow[13, 16, 17], and renormalization group flows=-=[6, 8, 12, 15]-=-, etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is the following coupled system ∂ ∂t g(x, t) = −2Ricg(x,t) + 4du(x, t) ⊗ du(x, t),(1.1) ∂ ∂t u(x, t) = ∆g(x,t)u(x... |

27 | Evolution of an extended Ricci flow system
- List
- 2005
(Show Context)
Citation Context ...hrinker entropys 35 References 43 1. Introduction After successfully applying the Ricci flow to topological and geometric problems, people study some analogues flows, including the harmonic-Ricci flow=-=[9, 11]-=-, connection Ricci flow[14], Ricci-Yang-Mills flow[13, 16, 17], and renormalization group flows[6, 8, 12, 15], etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is t... |

22 |
Eigenvalues and energy functionals with monotonicity formulae under Ricci flow,
- Li
- 2007
(Show Context)
Citation Context ...t)(1.14) + 1 4 ∫ M ∣∣Sg(t),u(t) + 4du(t) ⊗ du(t)∣∣2g(t) e−ϕ(t)dVg(t) − ∫ M ∆g(t) (∣∣∣g(t)∇u(t)∣∣∣2 g(t) ) e−ϕ(t)dVg(t) where f(t)2 = e−ϕ(t). Remark 1.14. When u ≡ 0, (1.14) reduces to J. Li’s formula =-=[7]-=-. Suppose thatM is a closed manifold of dimension n. For any Riemannian metric g, any smooth functions u, f , and any positive number τ , we define (1.15) W±(g, u, f, τ) := ∫ M [ τ ( Sg + | g∇f |2g ) ... |

20 | On second variation of Perelman’s Ricci shrinker entropy - Cao, Zhu |

15 |
Eigenvalues of (−∆ + R ) on manifolds with nonnegative curvature operator”,
- Cao
- 2007
(Show Context)
Citation Context ...1. If Ricg(0) − 2du(0) ⊗ du(0) ≥ 0, then the eigenvalues of the operator ∆g(t),u(t) are nondecreasing under the harmonic-Ricci flow. Remark 1.12. If we choose u(t) ≡ 0, then we obtain X. Cao’s result =-=[3]-=-. There is another expression of ddtλ(t). Theorem 1.13. Suppose that (g(t), u(t)) is a solution of the harmonicRicci flow on a compact Riemannian manifold M and f(t) is an eigenvalue of ∆g(t),u(t), i.... |

11 |
First eigenvalue of geometric operators under the Ricci flow.
- Cao
- 2008
(Show Context)
Citation Context ...the harmonic-Ricci flow Recall (8.1) µ(g, u) = µ1(g, u) = inf { F(g, u, f) ∣∣∣ ∫ M e−fdVg = 1 } . We showed that µ(g, u) is the smallest eigenvalue of the operator −4∆g + Rg − 2 | g∇u|2g. Inspired by =-=[3, 4]-=-, we define a Laplacian-type operators associated with quantities g, u, c: ∆g,u,c := −∆g + c ( Rg − 2 | g∇u|2g ) ,(8.2) ∆g,u := ∆g,u, 1 2 = −∆g + 1 2 ( Rg − 2 | g∇u|2g ) .(8.3) Then µ(g, u) is the sma... |

10 | Monotone volume formulas for geometric flows - Müller |

8 |
Regularity and expanding entropy for connection Ricci flow
- Streets
(Show Context)
Citation Context ... 43 1. Introduction After successfully applying the Ricci flow to topological and geometric problems, people study some analogues flows, including the harmonic-Ricci flow[9, 11], connection Ricci flow=-=[14]-=-, Ricci-Yang-Mills flow[13, 16, 17], and renormalization group flows[6, 8, 12, 15], etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is the following coupled system... |

6 |
Ricci flow coupled with harmonic map flow,
- Muller
- 2012
(Show Context)
Citation Context ...hrinker entropys 35 References 43 1. Introduction After successfully applying the Ricci flow to topological and geometric problems, people study some analogues flows, including the harmonic-Ricci flow=-=[9, 11]-=-, connection Ricci flow[14], Ricci-Yang-Mills flow[13, 16, 17], and renormalization group flows[6, 8, 12, 15], etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is t... |

6 |
Ricci Yang–Mills Flow
- Streets
- 2007
(Show Context)
Citation Context ...uccessfully applying the Ricci flow to topological and geometric problems, people study some analogues flows, including the harmonic-Ricci flow[9, 11], connection Ricci flow[14], Ricci-Yang-Mills flow=-=[13, 16, 17]-=-, and renormalization group flows[6, 8, 12, 15], etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is the following coupled system ∂ ∂t g(x, t) = −2Ricg(x,t) + 4du(x... |

6 | Ricci Yang-Mills flow on surfaces,
- Streets
- 2010
(Show Context)
Citation Context ...uccessfully applying the Ricci flow to topological and geometric problems, people study some analogues flows, including the harmonic-Ricci flow[9, 11], connection Ricci flow[14], Ricci-Yang-Mills flow=-=[13, 16, 17]-=-, and renormalization group flows[6, 8, 12, 15], etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is the following coupled system ∂ ∂t g(x, t) = −2Ricg(x,t) + 4du(x... |

6 |
Modified Ricci Flow on a Principal Bundle
- Young
- 2008
(Show Context)
Citation Context ...uccessfully applying the Ricci flow to topological and geometric problems, people study some analogues flows, including the harmonic-Ricci flow[9, 11], connection Ricci flow[14], Ricci-Yang-Mills flow=-=[13, 16, 17]-=-, and renormalization group flows[6, 8, 12, 15], etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is the following coupled system ∂ ∂t g(x, t) = −2Ricg(x,t) + 4du(x... |

3 | Generalized Ricci flow I: higher derivatives estimates for compact manifolds, arXiv: math.DG/0905.0045 - Li |

3 | Singularity of renormalization group flows
- Streets
(Show Context)
Citation Context ...gical and geometric problems, people study some analogues flows, including the harmonic-Ricci flow[9, 11], connection Ricci flow[14], Ricci-Yang-Mills flow[13, 16, 17], and renormalization group flows=-=[6, 8, 12, 15]-=-, etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is the following coupled system ∂ ∂t g(x, t) = −2Ricg(x,t) + 4du(x, t) ⊗ du(x, t),(1.1) ∂ ∂t u(x, t) = ∆g(x,t)u(x... |

2 |
Generalized Ricci flow I: Local existence and uniqueness, Topology and physics
- He, Hu, et al.
- 2008
(Show Context)
Citation Context ...gical and geometric problems, people study some analogues flows, including the harmonic-Ricci flow[9, 11], connection Ricci flow[14], Ricci-Yang-Mills flow[13, 16, 17], and renormalization group flows=-=[6, 8, 12, 15]-=-, etc. In this note, we study the eigenvalue problems of the harmonic-Ricci flow which is the following coupled system ∂ ∂t g(x, t) = −2Ricg(x,t) + 4du(x, t) ⊗ du(x, t),(1.1) ∂ ∂t u(x, t) = ∆g(x,t)u(x... |

2 | The second variation of the Ricci expanding entropy - Zhu |