Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones polynomials.

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We prove a second fundamental theorem in the sense of Nevanlinna's theory of meromorphic functions replacing the constants$a$in $$\bar N$$ ($r, f, a$) by rational functions$R$with$R(∞)=a$. The key argument is Ahlfors's second fundamental theorem from his theory of covering surfaces.

In [$Rong$, F., Quasi-parabolic analytic transformations of$C$^{$n$},$J. Math. Anal. Appl.$$343$(2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of$C$^{$n$}. Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.

Delaunay meshes (DM) are a special type of manifold triangle meshes --- where the local Delaunay condition holds everywhere --- and find important applications in digital geometry processing. This paper addresses the \yongjin{general} DM simplification problem: given an arbitrary manifold triangle mesh M with n vertices and the user-specified resolution m(< n), compute a Delaunay mesh M* with m vertices that has the least Hausdorff distance to M. To solve the problem, we abstract the simplification process using a 2D Cartesian grid model, in which each grid point corresponds to triangle meshes with a certain number of vertices and a simplification process is a monotonic path on the grid. We develop a novel differential-evolution-based method to compute a low-cost path, which leads to a high quality Delaunay mesh. Extensive evaluation shows that our method consistently outperforms the existing methods in terms of approximation error. In particular, our method is highly effective for small-scale CAD models and man-made objects with sharp features but less details. Moreover, our method is fully automatic and can preserve sharp features well and deal with models with multiple components, whereas the existing methods often fail.