Study on Geometry and Analysis of Conformal Manifolds and Bubbling Trees

共形流形和冒泡树的几何与分析研究

基本信息

批准号：
14540072
负责人：
AKUTAGAWA Kazuo
金额：
$ 2.3万
依托单位：
Shizuoka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540072/
关键词：
Conformal Geometry Yamabe Invariant Scalar Curvature Cylindrical Manifolds Index Theorem Mass Invariant Inverse Mean Curvature Flow Technique Non-linear Analysis オービフォールドワイル不変量

项目摘要

We have studied the following:1.Study of cylindrical and orbifold Yamabe invariantsAs a generalization of the Yamabe constant/invariant of closed manifolds, we defined appropriately the orbifold Yamabe constant/invariant in terms of the cylindrical Yamabe constant/invariant.For a cylindrical 4-manifold with positive cylindrical Yamabe invariant, we also established a method for estimating its cylindrical Yamabe invariant from above, by means of the Atiyah-Patodi-Singer L^2-index theory. Moreover, we generalized the Kobayashi inequality for Yamabe invariants to cylindrical Yamabe invariants, and studied its applications.2.Study on the mass of compact conformal manifoldsThe mass is a geometric invariant for asymptotically flat manifolds. For a compact conformal manifold (M, C) with positive Yamabe invariant, a scalar-flat, asymptotically flat manifold (M-{p},g_<AF>) is defined naturally from each initial metric g in C, where [g_<AF>]=C. Then the mass m(g ; p) is non-negative. This mass m … More (g ; p) also depends on the choice of g and p. However, if we use the Habermann-Jost's canonical metric g_<HJ> as a initial metric, then the mass m(g_<HJ>;p) is now independent of the choice of p. By using this fact, we can define the mass mass(M ; C) of the conformal manifold (M, C) as a conformal invariant. Moreover, taking the infimum of it over all conformal classes, we can also define the mass invariant mass(M) as a differential-topological invariant of M. We studied on the Kobayashi-type inequality of the mass invariant for connected manifolds.3.Yamabe invariants of 3-manifoldsThe method of inverse mean curvature flow is the central technique for the resolution of the Riemannian Penrose Conjecture in Cosmology. By using this technique, Bray-Neves determined the value of the Yamabe invariant of RP^3. This result is the first affirmative answer to the Schoen's Conjecture for the Yamabe invariant of 3-manifolds with constant curvature. We also determined the Yamabe invariant of the connected manifold RP^3 # k(S^2 x S^1), by means of the inverse mean curvature flow technique. This is also one of the open problems proposed by Bray-Neves.For the above study, the support by the 'Grant-in-Aid for Sci. Res. (C)(2),14540072' was very important. Less

We have studied the following:1.Study of cylindrical and orbifold Yamabe invariantsAs a generalization of the Yamabe constant/invariant of closed manifolds, we defined appropriately the orbifold Yamabe constant/invariant in terms of the cylindrical Yamabe constant/invariant.For a cylindrical 4-manifold with positive cylindrical Yamabe invariant, we还建立了一种通过atiyah-patodi-singer l^2 index理论来估计其Yamabe不变的方法。此外，我们将Yamabe不变性的Kobayashi不平等概括为Yamabe不变性的，并研究了其应用。2。在紧凑的结构歧管质量上进行研究，质量是非对称平面歧管的几何不变。对于具有阳性不变的正的紧凑共形歧管（m，c），标量液量，不对称的扁平歧管（m- {p}，g_ <af>）是自然定义的。那么质量m（g; p）是非负的。这个质量m…更多（g; p）也取决于g和p的选择。但是，如果我们将Habermann-Jost的规范度量G_ <HJ>用作初始度量，则质量M（G_ <HJ>; P）现在独立于p的选择。通过使用这个事实，我们可以将保形歧管（M，C）的质量（M; C）定义为保形不变。此外，将其限制在所有子同类阶级上，我们还可以将质量不变的质量（M）定义为M的差异性。在宇宙学中。通过使用此技术，Bray-neves确定了Rp^3的Yamabe不变的值。该结果是对Schoen对3个manifolds的Yamabe不变的稳定质量不变的猜想的第一个肯定答案。我们还通过逆平均曲率流动技术确定了连接的歧管RP^3＃K（S^2 x s^1）的Yamabe不变性。对于上述研究，这也是Bray-neves提出的开放问题之一。 res。（c）（2），14540072'非常重要。较少的