Citation Context

....e. theorems stating that harmonic maps in a certain class of maps and under certain topological or geometric constraints are necessarily constant [33]. Due to Perelman’s proof of the geometrization and hence the Poincare conjecture using Ricci flow [25,27,26], there is now a strong interest in studying manifolds M with time-dependent geometry. In such a context, the notion of harmonic map turns out to be no longer appropriate; however, it is natural to study space–time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. [18,23,35]. The behavior of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5,21,22]. It is clear from the static case that in the nonlinear situation under Ricci flow also the geometry of the target space will naturally play a crucial role, see e.g. [19,31]. Building on our previous work on martingales on manifolds with time-dependent connection [10], we establish stochastic representation formulae for space–time harmonic maps and solutions to the harmonic map heat flow defined on a manifold with time-dependent metric. We then...

Citation Context

...out to be no longer appropriate; however, it is natural to study space-time harmonic maps which by time reversal provide solutions to the harmonic map heat flow (or nonlinear heat equation), see e.g. =-=[19, 24, 36]-=-. The behaviour of (positive) solutions to the linear heat equation under Ricci flow has been intensively studied during the last decade, e.g. [5, 22, 23]. It is clear from the static case that in the...

Citation Context

...ows. In this direction, many fundamental theorems in Ricci flows are extended to Ricci-harmonic flows, for example, no breather theorems, non-collapsing theorems [14, 16], Perelman’s entropy formulas =-=[10]-=-, Monotone volume formulas [15], and volume growth estimates [19]. See also [18] for more related results. In this short note, we give lower diameter bounds for closed domain manifolds of shrinking Ri...

Citation Context

...f(t) = −∆g(t)f(t) + ∣∣∇g(t)f(t)∣∣2g(t) −Rg(t) + m2τ(t) ,(1.6) d dt τ(t) = −1; Thus, (1.3) is a mixture of (1.5) and (1.6). There also are lots of interesting generalized Ricci flows, for example, see =-=[14, 17, 18, 19, 20, 21]-=-. 4 YI LI 1.3. A parabolic flow. In this paper, we consider a class of Ricc flow type parabolic differential equation: ∂tg(t) = −2Ricg(t) + 2α1∇g(t)φ(t)⊗∇g(t)φ(t) + 2α2∇2g(t)φ(t),(1.7) ∂tφ(t) = ∆g(t)φ...