Citation Context

... Ricci flow presented in [12]. Some other examples are given in the second section of this article, the main examples and motivation for this work being List’s extended Ricci flow system developed in =-=[8]-=-, the Ricci flow coupled with harmonic map heat flow presented in [11], and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monoto...

Citation Context

...an also study the coupling of these two flows as a subject of independent interest. The case where the target manifold for the harmonic map flow is a subset of the real line was considered by List in =-=[12]-=-. This coupling has relevance to general relativity and solutions of the Einstein equations. The case of arbitrary target manifolds was subsequently studied by Müller in [17] under the name (RH)α flo...

Citation Context

...ime if the metric has a static or perhaps even a stationary Killing vector field (recall that the Ricci flow preserves isometries). 1 A geometric flow of static Lorentzian metrics was studied by List =-=[15]-=-. He did not begin with the Ricci flow of a static metric (i.e., a metric with timelike, hypersurface-orthogonal Killing field). Instead he presented a system of flow equations whose fixed points solv...

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...

Citation Context

...∇u,∇u) = △S + 2 〈∇u,∇△u〉+ 2|△u|2 + 2|Rc|2 − 2Rc(∇u,∇u). Combining equations above yields (2.9) ∂ ∂t S = △S + 2|△u|2 + 2|Sij |2. Remark 2.4. System (2.6) and some evolution equations above appeared in =-=[13]-=- with a constant αn associated with the term du⊗du. However, in case αn ≥ 0 if letting ũ = √αnu recovers (2.6). So every result in section 4 holds for αn ≥ 0 as well. Remark 2.5. A generalization of ...

Citation Context

...∞ max x∈Xi0 (t) |∇̂u|ĝ(t)(x) = 0. For all sufficiently small i0, if t is sufficiently large then Xi0(t) is nonempty. Remark 1.2. The proof of Theorem 1.1.(i) is essentially contained in List’s paper =-=[Lis08]-=- on a modified Ricci flow. The only thing missing from [Lis08] is the observation that his flow differs from the warped product flow by a Lie derivative. Remark 1.3. The proof of the curvature bound i...

Citation Context

...ral relativity. List has proposed a geometric flow from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations =-=[19]-=-. In this note, we mainly generalize List’s results to the case with negative cosmological constant. In this case, the spacetime (Xn+1, h̃) satisfies the equation (1) G+ Λh̃ = 8πT where G = Ric(h̃) − ...