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...n of a proof of the Poincare conjecture followed. Some of these are collected in [2]. The recent approach of Perelman to prove the full geometrization conjecture added new ideas to Hamilton’s program =-=[6, 7]-=-. In particular, Perelman showed that the Ricci flow can be interpreted as a gradient flow of an entropy functional and thus that the canonical metrics are stationary points of this entropy in the var...

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...k from problems in low-dimensional topology and geometry to physical questions in general relativity. 1. Introduction In the last 20 years, the Ricci flow system for a Riemannian metric introduced in =-=[1]-=- ∂tg = −2Rc(g) has been used with great success for the construction of canonical metrics on Riemannian manifolds of low dimension. In his first paper on the Ricci flow, Hamilton proved that given an ...

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...ysical property of an asymptotically flat manifold is its ADM mass. For manifolds as defined above, it is given by the coefficient m in the expansion of (g, u). The general definition can be found in =-=[25]-=- where it is also shown that the ADM mass is invariantly defined and independent of the asymptotic coordinate system. We prove that the flow (3.1) preserves the class of asymptotically flat manifolds ...

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...em (3.1) is weakly parabolic. Therefore we can use DeTurck’s method [16] to prove short time existence and uniqueness on closed manifolds. For the complete case we show [15, §3] that the proof of Shi =-=[17]-=- can be modified to the situation at hand. This yields the following general existence theorem: Theorem 4.1. Let (Σ, g̃) be a smooth complete noncompact n-dimensional Riemannian manifold with |R̃m|20 ...

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...s the result follows from a straightforward calculation; see [15, Lemma 2.13]. 4. Short time existence As the Ricci flow, the system (3.1) is weakly parabolic. Therefore we can use DeTurck’s method =-=[16]-=- to prove short time existence and uniqueness on closed manifolds. For the complete case we show [15, §3] that the proof of Shi [17] can be modified to the situation at hand. This yields the following...

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...on of the original weakly parabolic system (3.1) together with the stated estimates [15, Theorem 3.22]. Remark 4.2. We do not prove uniqueness of the solution in Theorem 4.1. However, the result in =-=[18]-=- strongly suggests that solutions satisfying the above bounds are unique. In particular [19, Appendix B.3] applies to (3.1) on complete and asymptotically flat manifolds. 5. A priori estimates and lon...

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... ∫ M (4πτ)−n/2e−fdV = 1 } . Proposition 6.4. Let M be closed and connected. Then μ is attained by a smooth function f̄ ∈ C∞(M) satisfying the normalization constraint. Proof. We adapt the method from =-=[23]-=- to our situation and refer to the proof of [15, Proposition 5.8] for the technical details. Remark 6.5. W is still bounded below on complete Σ, but a weakly convergent minimizing sequence in W 1,2 ...

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...n of a proof of the Poincare conjecture followed. Some of these are collected in [2]. The recent approach of Perelman to prove the full geometrization conjecture added new ideas to Hamilton’s program =-=[6, 7]-=-. In particular, Perelman showed that the Ricci flow can be interpreted as a gradient flow of an entropy functional and thus that the canonical metrics are stationary points of this entropy in the var...

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...ting candidate for such a smoothing operator. It should be possible to approximate static solutions by solutions to (1.1). Another application relates to the quasi-local mass definition of Bartnik in =-=[10]-=- and the recent paper of [11]. After adding suitable parabolic boundary conditions, our system could be helpful in the construction of the static minimal mass extension which would provide the minimum...

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...othing operator. It should be possible to approximate static solutions by solutions to (1.1). Another application relates to the quasi-local mass definition of Bartnik in [10] and the recent paper of =-=[11]-=-. After adding suitable parabolic boundary conditions, our system could be helpful in the construction of the static minimal mass extension which would provide the minimum in the definition of this ma...

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...a short introduction to static vacuum solutions of the Einstein equations. Further material can be found in [12, chapter 2] and [13, chapter 6]. A very thorough and detailed discussion is provided in =-=[14]-=-. From a geometrical point of view, a Lorentzian manifold (L4, h) is said to be static if there exists a 1-parameter group of isometries with timelike orbits and a spacelike hypersurface Σ which is or...

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...singular time and close enough to the singular point with the rescaling limit is possible. It is crucial to know what these regions look like to set up the delicate surgery procedures as described in =-=[5]-=- or [7]. We first give some definitions. Definition 7.7. A solution (g, u)(t) to (3.1) on a complete Riemannian manifold is called ancient, if it exists for all t ∈ (−∞, T ] up to some time T ≥ 0. Def...