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Geometric and topological rigidity theorem for 3-manifolds

三流形的几何和拓扑刚性定理

基本信息

批准号：
12640092
负责人：
SOMA Teruhiko
金额：
$ 2.24万
依托单位：
Tokyo Denki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

项目摘要

The researcher has studied thoroughly geometric and topological rigidity theorems for 3-manifolds. In particular, he found out that the existence of least area planes properly embedded in the universal coverings in the proof of topological rigidity theorems. Let M be a closed hyperbolic 3-manifold and p:H^3→M the universal covering. Here, we suppose that M has a Riemannian metric which is not necessarily hyperbolic. The metric r on H^3 induced from that on M is called a co-compact metric. D.Gabai conjectured that "any simple smooth curve in the boundary S^2_∞ of H^3 spans a properly embedded r-least area plane in H^3"(J.Amer.Math.Soc.10(1997)). Throughout this project, the researcher proved that the conjecture is true. Moreover, he proved that the result holds when π_1(M) is Gromov-hyperbolic even if M is not a hyperbolic 3-manifold. That is, it was shown that, for the universal converging M^^〜 of the manifold M, any Jordan curve in ∂M^^〜 bounds a properly embedded r-least area plane in M.Furthermore, the researcher solved the question "What kinds of topological types do geometric limits of quasi-Fuchsian groups have ?" completely. Precisely, Σis a closed orientable surface of genus>1, and {p_n} is an algebraically convergent sequence of quasi-Fuchsian representations ρ_n:π_1(Σ)→PSL_2(C). Suppose that the sequence {Γ_n} consisting of the quasi-Fuchsian groups Γ_n=ρ_n(π_1(Σ)) converges geometrically to a Kleinian group G. Then, the researcher proved that there exists a closed set Χ in Σ×[0,1] called a crevasse so that H^3/G is homeomorphic to Σ×[0,1]-Χ. Conversely, it was also proved that, for any crevasse Χ in Σ×[0,1], there exists a geometric, limits G of quasi-Fuchsian groups such that H^3/G is homeomorphic to Σ×[0,1]-Χ.

研究人员对三维流形的几何刚性和拓扑刚性定理进行了深入的研究。特别地，在证明拓扑刚性定理时，他发现了最小面积平面的存在恰好嵌入在泛覆盖中。设M是闭双曲三维流形，p：H^3→M是泛覆盖。这里，我们假设M有一个黎曼度量，它不一定是双曲的。由M上的度量导出的H^3上的度量r称为余紧度量。D.Gabai猜想“在H^3的边界S^2_∞上的任何简单光滑曲线都横跨在H^3中的适当嵌入的r-最小面积平面上”(J.amer.Math.Soc.10(1997年))。在整个项目中，研究人员证明了这个猜想是正确的。此外，他还证明了当π_1(M)是Gromov-双曲流形时，即使M不是双曲三维流形，结果仍然成立。也就是说，对于流形M的泛收敛的M^^~，∂M^~中的任一Jordan曲线都限定于M中适当嵌入的r-最小面积平面。此外，作者还解决了“拟Fuchsian群的几何极限具有哪种拓扑类型？”完全地。确切地说，Σ是亏格&gt；1的闭可定向曲面，且{p_n}是拟富克斯表示的代数收敛序列ρ_n：π_1(Σ)→pSL_2(C).设由拟Γ群Γ_n=ρ_n(π_1(Σ))组成的序列{ρ_n}几何收敛于Klein群G，则证明了在Σ×[0，1]中存在称为裂缝的闭集Σ，使得H^3/G同胚于Χ×[0，1]-Kleinan群G。反之，还证明了对于Χ×[0，1]中的任一裂隙Σ，存在拟Fuchsian群的几何极限G，使得H^3/G同胚于Σ×[0，1]-Χ.