## Volume change under drilling.(English)Zbl 1031.57014

Given a complete hyperbolic 3-manifold M containing an embedded tubular $$R$$-neighborhood of closed geodesic $$\gamma$$, the author estimates the volume of a complete hyperbolic metric on the complement $$M_{\gamma}=M\backslash\gamma$$: $\text{Vol}(M_{\gamma})\leq (\coth R)^{1/2}(\coth 2R)^{1/2}\text{Vol}(M) .$ As a corollary he shows that the smallest volume orientable hyperbolic 3-manifold has volume $$>0.32$$.

### MSC:

 57M50 General geometric structures on low-dimensional manifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C22 Geodesics in global differential geometry

### Keywords:

Hyperbolic structure; 3-manifold; volume; geodesic

SnapPea
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### References:

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