论文标题
重新归一化体积和3维凸内核的Weil-Petersson梯度流
The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores
论文作者
论文摘要
在本文中，我们使用Weil-Petersson梯度流进行重新归一化的体积来研究相对酰基三个manifold $（n; s）$的凸Cocompact双曲线结构的空间$ cc（n; s，x）$。感兴趣的情况下是酰基歧管的变形空间，而与固定表面相关的准富奇西亚空间的晶圆切片。为了治疗沿流线变性的可能性，我们引入了手术程序，以产生一个手术梯度流，该梯度流量限制在CC（N; S，X）中的独特结构$ M _ {\ rm geod} \中，并具有完全GeoDesicic convex convex Corex Core convex Core convex Core conevex Core Barge fac $ s $ s $ s $ s $。分析沿流线的结构的几何形状，我们表明，如果$ v_r（m）$是$ m $的重量化量，则$ v_r（m）-v_r（m _ {\ rm geod}）$在下面由Weil-petersson discome $ d _ {\ rm wp}的线性函数限制在下面M _ {\ rm geod}）$，仅取决于$ s $的拓扑。手术流在研究双曲线3个字体研究中为许多问题提供了一种统一的方法，从而提供了诸如Storm的众所周知定理的新证明和概括，即$ M _ {\ rm geod} $具有$ N $ N $ N $ N $ nindrical的最小量和第二作者的结果比较Convex Core ands Secords和Weil-Peterssssssssssson的量。
In this paper, we use the Weil-Petersson gradient flow for renormalized volume to study the space $CC(N;S,X)$ of convex cocompact hyperbolic structures on the relatively acylindrical 3-manifold $(N;S)$. Among the cases of interest are the deformation space of an acylindrical manifold and the Bers slice of quasi-Fuchsian space associated to a fixed surface. To treat the possibility of degeneration along flow-lines to peripherally cusped structures, we introduce a surgery procedure to yield a surgered gradient flow that limits to the unique structure $M_{\rm geod} \in CC(N;S,X)$ with totally geodesic convex core boundary facing $S$. Analyzing the geometry of structures along a flow line, we show that if $V_R(M)$ is the renormalized volume of $M$, then $V_R(M)-V_R(M_{\rm geod})$ is bounded below by a linear function of the Weil-Petersson distance $d_{\rm WP}(\partial_c M, \partial_c M_{\rm geod})$, with constants depending only on the topology of $S$. The surgered flow gives a unified approach to a number of problems in the study of hyperbolic 3-manifolds, providing new proofs and generalizations of well-known theorems such as Storm's result that $M_{\rm geod}$ has minimal volume for $N$ acylindrical and the second author's result comparing convex core volume and Weil-Petersson distance for quasifuchsian manifolds.
