Monotone volume formulas for geometric flows

Abstract

We consider a closed manifold M with a Riemannian metric gij (t) evolving by ∂t gij = −2Sij where Sij (t) is a symmetric two-tensor on (M, g(t)). We prove that if Sij satisfies the tensor inequality D(Sij, X) ≥ 0 for all vector fields X on M , where D(Sij, X) is defined in (1.6), then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the flow ∂t gij = −2Sij . In the case where Sij = Rij , the Ricci curvature of M , the result corresponds to Perelman's well-known reduced volume monotonicity for the 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 monotonicity formulas for these flows.

Ricci flow coupled with harmonic map flow

“…This follows from our more general result from [29]. We apply this monotonicity to deduce a local non-collapsing theorem.…”

“…The proof follows Perelman's results for the Ricci flow [31] very closely, see also the notes on his paper by Kleiner and Lott [19] and the book by Morgan and Tian [25]. However, we need the more general results from [29] that also hold for our coupled flow system. We only give a sketch.…”

“…Together with Ω(τ ′ ) ⊂ Ω(τ ) for τ ≤ τ ′ this yields an alternative proof of the monotonicity of the reduced volumes obtained in [29]. Moreover, it implies that the contribution to the reduced volumeṼ k (ε k r Claim 2:Ṽ k (t k ) is bounded below away from zero.…”

“…Let us briefly restate the main result from our previous article [29] about the monotonicity of reduced volumes for flows of the form ∂ ∂t g ij = −2S ij , where S ij is a symmetric tensor with trace S = g ij S ij . (The resuts can also be found in the authors thesis [28].…”

Optimal transportation and monotonic quantities on evolving manifolds

“…List [2008] considered an extended Ricci flow in his thesis, and generalized the monotonicity of Perelman's ᐃ-entropy to his flow. Müller [2010] studied more general evolving closed manifolds (M, g i j (t)) with metrics g i j (t) satisfying the equation…”