Monotone volume formulas for geometric flows

By Reto Müller. 2009

67 citations recorded by the source corpus.

Abstract We consider a closed manifold M with a Riemannian metric gij (t) evolving by ∂tgij = –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 ∂tgij = –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 [The entropy formula for the Ricci flow and its geometric applications, 2002]. 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 [Evolution of an extended Ricci flow system, AEI Potsdam, 2005], the Ricci flow coupled with harmonic map heat flow presented in [Müller, The Ricci flow coupled with harmonic map heat flow, ETH Zürich, 2009], and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monotonicity formulas for these flows.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/03b2bb818b86aa278a78ae92aa805d7dfbd8717d

How researchers see this paper

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In Applied Mathematics

1. Anton Thalmaier
2. Robert Philipowski
3. Hongxin Guo
4. Reto Muller
5. Feng-Yu Wang

In Astronomy and Astrophysics

1. Anton Thalmaier
2. Reto Muller
3. Robert Philipowski
4. Hongxin Guo

Citations in the corpus

• ENTROPY AND LOWEST EIGENVALUE ON EVOLVING MANIFOLDS (2013, background): “For instance, very nice general approaches to get monotone quantities on evolving manifolds have been developed in [4, 10].”
• Ricci flow coupled with harmonic map flow (2009, background): “Here, Lbexp τ1 xk is the Lb-exponential map defined in [29], taking v to η(τ1) with η being the Lb-geodesic with initial vector v, J(v, τ1) = det d(Lbexp τ1 xk) denotes the Jacobian of Lbexp τ1 xk and Ω(τ1, k) ⊂ TxkM is a set which is mapped bijectively to M up to a set of measure zero under the map”