Volume change under drilling

关于钻孔和填充的线性体积界

Linear volume bounds for drilling and filling

本文研究了双曲3流形上钻孔和填充操作的体积变化，并给出了线性界。

Bounds on Volume Increase under Dehn Drilling Operations

In this paper we investigate how the volume of hyperbolic manifolds increases under the process of removing a curve, that is, Dehn drilling. If the curve we remove is a geodesic we are able to show that for a certain family of manifolds the volume increase is bounded above by $π\cdot l$ where $l$ is the length of the geodesic drilled. Also we construct examples to show that there is no lower bound to the volume increase in terms of a linear function of a positive power of length and in particular volume increase is not bounded linearly in length.

$L^2$-界用于在凸余紧双曲3-流形中钻孔短测地线

$L^2$-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds

本文研究了凸余紧双曲3-流形钻孔短测地线后边界复射影结构的变化，给出了不依赖于边界单射半径的$L^2$-界，证明其变化受普适常数乘以测地线长度平方根的约束。

Effective drilling and filling of tame hyperbolic 3-manifolds

We give effective bilipschitz bounds on the change in metric between thick parts of a cusped hyperbolic 3-manifold and its long Dehn fillings. In the thin parts of the manifold, we give effective bounds on the change in complex length of a short closed geodesic. These results quantify the filling theorem of Brock and Bromberg, and extend previous results of the authors from finite volume hyperbolic 3-manifolds to any tame hyperbolic 3-manifold. To prove the main results, we assemble tools from Kleinian group theory into a template for transferring theorems about finite-volume manifolds into th…

Stability of Einstein metrics and effective hyperbolization in large Hempel distance

Extending earlier work of Tian, we show that if a manifold admits a metric that is almost hyperbolic in a suitable sense, then there exists an Einstein metric that is close to the given metric in the $C^{2,α}$-topology. In dimension $3$ the original manifold only needs to have finite volume, and the volume can be arbitrarily large. Applications include a new proof of the hyperbolization of $3$-manifolds of large Hempel distance yielding some new geometric control on the hyperbolic metric, and an analytic proof of Dehn filling and drilling that allows the filling and drilling of arbitrary many …

Effective bilipschitz bounds on drilling and filling

This paper proves explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic 3-manifold N and the thick part of any of its long Dehn fillings. Given a bilipschitz constant J > 1 and a thickness constant epsilon > 0, we quantify how long a Dehn filling suffices to guarantee a J-bilipschitz map on epsilon-thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff's theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infi…