Study on Geometry and Analysis of Conformal Manifolds and Bubbling Trees

我们研究了以下内容：1。作为封闭流形的Yamabe常数/不变量的推广，我们用柱面Yamabe常数/不变量适当地定义了轨道Yamabe常数/不变量。对于具有正柱面Yamabe不变量的柱面4流形，我们也利用Atiyah-Patodi-Singer L^2指标理论，建立了一种从上面估计柱面Yamabe不变量的方法。此外，我们将Yamabe不变量的Kobayashi不等式推广到圆柱形的Yamabe不变量，并研究了它的应用。紧形共形流形质量的研究质量是渐近平面流形的几何不变量。对于具有正Yamabe不变量的紧形共形流形（M， C），从C中的每个初始度量g自然地定义了一个标量平坦的渐近平坦流形（M-{p},g_<AF>），其中[g_<AF>]=C。那么质量m（g; p）是非负的。这个质量m…More （g;p）也取决于g和p的选择。然而，如果我们使用Habermann-Jost的规范度规g_<HJ>作为初始度规，那么质量m（g_<HJ>;p）现在与p的选择无关。利用这一事实，我们可以将保形流形（m， C）的质量mass（m； C）定义为保形不变量。此外，取其在所有共形类上的最小值，我们也可以将质量不变量mass(M)定义为M的微分拓扑不变量。3流形的Yamabe不变量平均曲率反流方法是解决宇宙学中黎曼彭罗斯猜想的核心技术。通过使用这种技术，Bray-Neves确定了RP^3的Yamabe不变量的值。这个结果是对常曲率3流形的Yamabe不变量的舍恩猜想的第一个肯定的回答。我们还通过逆平均曲率流技术确定了连通流形RP^ 3# k（S^2 x S^1）的Yamabe不变量。这也是Bray-Neves提出的开放性问题之一。对于上述研究，我们得到了“科学资助计划”的支持。Res. (C)(2),14540072'非常重要。少

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