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首页> 外文期刊>Annales Henri Poincare >The Ground State and the Long-Time Evolution in the CMC Einstein Flow
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The Ground State and the Long-Time Evolution in the CMC Einstein Flow

机译：CMC爱因斯坦流的基态和长期演化

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摘要

Let (g,K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y (Σ)). As noted by Fischer and Moncrief, the reduced volume V(k) = (-k/3)~3 Vol_(g(k))(Σ) is monotonically decreasing in the expanding direction and bounded below by V_(inf) =(-1/6 Y (Σ))~(3/2). Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g_i,K_i)} satisfying: (i)k_i = -3, (ii) Vi ↓ V_(inf),(iii) Q_0((g_i,K_i)) ≤ Λ, where Q0 is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow (g, K)(σ) =((-k/3)~2 g, (-k/3)K), where σ =-ln(-k) ∈ [a,∞). We prove that if ε_1 = ε_1((g,K )) ≤ Λ (where ε_1 = Q_0 + Q_1, is the sum of the zero and first order Bel–Robinson energies) the flow (g, K )(σ) persistently geometrizes the three-manifold Σ and the geometrization is the ground state if V ↓ Vinf .
机译：令（g，K）（k）为CMC（真空）爱因斯坦流经紧凑型三歧管Σ的山形不变量（Y（Σ））为正。如Fischer和Moncrief所指出的，减小的体积V（k）=（-k / 3）〜3 Vol_（g（k））（Σ）在扩展方向上单调减小，并在下面以V_（inf）=（ -1/6 Y（Σ））〜（3/2）。受这一事实的启发，我们将流形Σ的基态定义为满足以下条件的任何CMC状态序列{（g_i，K_i）}的“极限”：（i）k_i = -3，（ii）Vi↓V_（inf） ，（iii）Q_0（（g_i，K_i））≤Λ，其中Q0是Bel–Robinson能量，而Λ是任意正常数。我们证明（作为几何状态）基态等效于Thurston几何化Σ。基态自然分为三类。我们为每个班级提供示例，包括我们将详细分析的新基态（Double Cusp）。最后，考虑长时间的宇宙学归一化流量（g，K）（σ）=（（-k / 3）〜2 g，（-k / 3）K），其中σ= -ln（-k）∈[ a，∞）。我们证明如果ε_1=ε_1（（g，K））≤Λ（其中ε_1= Q_0 + Q_1是零级和一阶Bel–Robinson能量之和），则流量（g，K）（σ）会持续几何化如果V↓Vinf，则三流形Σ和几何化为基态。

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