RUI:Hyperbolic 3-Manifolds and Knots

1978年，William Thurston通过证明许多3-流形具有双曲度量，彻底改变了低维拓扑。Mostow刚性定理则暗示这些度量可以用来产生有趣的不变量来区分双曲三维流形。 这些想法已经产生了大量的结果。这个命题的目的是为了进一步理解双曲三维流形。特别地，在双曲三维流形的尖点和双曲纽结中有一组不变量，包括尖点体积、子午线长度和经度长度。我们的计划是继续研究这些不变量，更好地描绘它们可能的取值范围，并确定它们对双曲结构其余部分的影响。此外，最近的工作证明了双曲纽结补中全测地曲面的存在性。他们的存在令人惊讶和兴奋。人们希望能够确定哪些节点拥有这样的曲面。理解双曲三维流形对于确定空间宇宙的形状有着重要意义。 这些不变量将是宇宙学家可以用来经验地识别宇宙的那种。虽然最近对宇宙背景辐射的测量表明，宇宙的曲率非常接近于平坦，但这可能意味着宇宙是双曲线的，但非常大。那么就需要更精细的双曲不变量来确定它的拓扑形状。

In 1978, William Thurston revolutionized low-dimensional topology bydemonstrating that many 3-manifolds have hyperbolic metrics. The MostowRigidity Theorem then implies that these metrics can be used to generateinteresting invariants to distinguish hyperbolic 3-manifolds. A tremendousnumber of results have come out of these ideas. The goal of this proposalis to further understand hyperbolic 3-manifolds. In particular, there are aset of invariants that come out of the cusps of hyperbolic 3-manifolds, andhyperbolic knots, including the cusp volume, the meridian length and thelongitude length. The plan is to continue to work on these invariants,delineating better their possible range of values and determining theirimplications for the rest of the hyperbolic structure. Moreover, recentwork has demonstrated the existence of totally geodesic surfaces inhyperbolic knot complements. Their existence is suprising and exciting. Itis hoped that a determination can be made as to which knots possess suchsurfaces.Understanding hyperbolic 3-manifolds has implications for determining theshape of the spatial universe. These invariants will be the kind thatcosmologists may use to empirically identify the universe. Although recentmeasurements of the cosmic background radiation point to a curvature veryclose to flat, this could mean that the universe is hyperbolic, but verylarge. Then more subtle hyperbolic invariants will be necessary todetermine its topological shape.