Mathematical Research Letters

Volume 19 (2012)

Number 4

A mass-decreasing flow in dimension three

Pages: 927 – 938



Robert Haslhofer (Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)


In this paper, we introduce a mass-decreasing flow for asymptotically flat three-manifolds with nonnegative scalar curvature. This flow is defined by iterating a suitable Ricci flow with surgery and conformal rescalings and has a number of nice properties. In particular, wormholes pinch off and nontrivial spherical space forms bubble off in finite time. Moreover, a noncompact variant of the Perelman-energy is monotone along the flow. Assuming a certain inequality between the mass and this Perelman-energy a priori, we can prove that the flow squeezes out all the initial mass.

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