Yamabe metrics on cylindrical manifolds

Yamabe metrics on cylindrical manifolds
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圆柱流形上的 Yamabe 度量

DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
B. Botvinnik
B. Botvinnik
中科院分区：
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文献类型：
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作者：
K. Akutagawa;B. Botvinnik

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抽象的。我们研究了一类特殊的开放流形。在……类别中 黎曼流形这些是具有柱面的完备流形 结束了。我们给出了这样的共形几何的一个自然设置 包括柱面Yamabe的适当概念的流形 常量/不变量。这导致了相应版本的山部车 圆柱流形上的问题。我们找到了一个积极的解决方案 这个Yamabe问题：我们证明了最小化度量的存在性 并分析它们近乎无穷远的奇点。这些奇点转变为 具有非常特殊的类型：或者几乎是圆锥的，或者是几乎是尖峰的。我们描述了上确界的情况，即当圆柱形 Yamabe常数等于球面的Yamabe不变量。 我们证明了在这种情况下，这样的圆柱流形共形重合 当标准球体在有限数量的 积分。在研究这个最优案例的过程中，我们建立了一个 具体的渐近平坦流形的正质量定理 两个近乎圆锥的奇点。作为副产品，我们重温已知的 手术结果和Yamabe不变量。
Abstract. We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding version of the Yamabe problem on cylindrical manifolds. We find a positive solution to this Yamabe problem: we prove the existence of minimizing metrics and analyze their singularities near infinity. These singularities turn out to be of very particular type: either almost conical or almost cuspsingularities. We describe the supremum case, i.e., when the cylindrical Yamabe constant is equal to the Yamabe invariant of the sphere. We prove that in this case such a cylindrical manifold coincides conformally with the standard sphere punctured at a finite number of points. In the course of studying the supremum case, we establish a Positive Mass Theorem for specific asymptotically flat manifolds with two almost conical singularities. As a by-product, we revisit known results on surgery and the Yamabe invariant.