最小体积的闭曲面丛
Closed surface bundles of least volume
摘要：
由于双曲3-流形的体积集是良序的，对于每个固定的g，在极小体积圆上存在亏格-g曲面丛。在这里，我们引入一个明确的家庭的genus-g丛，我们猜想是唯一的这样的流形的最小体积。有条件的一个非常合理的假设，我们证明，这确实是当g是大的情况。证明结合了一个软几何极限参数与详细的Neumann-Zagier渐近公式的德恩填充量。我们的例子都是在Whitehead流形的兄弟上的Dehn填充，我们还分析了以这种方式获得的所有闭曲面丛的扩张，确定了那些具有最小扩张的。这给出了新的家庭的pseudo-Anosovs低膨胀，包括属7的例子，其中最大限度地减少膨胀之间的所有定向不变叶理。
英文摘要：
Since the set of volumes of hyperbolic 3-manifolds is well ordered, for each fixed g there is a genus-g surface bundle over the circle of minimal volume. Here, we introduce an explicit family of genus-g bundles which we conjecture are the unique such manifolds of minimal volume. Conditional on a very plausible assumption, we prove that this is indeed the case when g is large. The proof combines a soft geometric limit argument with a detailed Neumann-Zagier asymptotic formula for the volumes of Dehn fillings. Our examples are all Dehn fillings on the sibling of the Whitehead manifold, and we also analyze the dilatations of all closed surface bundles obtained in this way, identifying those with minimal dilatation. This gives new families of pseudo-Anosovs with low dilatation, including a genus 7 example which minimizes dilatation among all those with orientable invariant foliations.
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