arXiv subjects
Publications and source records attributed to Jesús Gonzalo Pérez.
We present two proofs that all closed, orientable 3-manifolds are parallelisable. Both are based on the Lickorish-Wallace surgery presentation; one proof uses a refinement due to Kaplan and some basic contact geometry. This complements a recent paper by Benedetti-Lisca.
For any closed Riemannian manifold $X$ we prove that large isoperimetric regions in $X\times{\mathbb R}^n$ are of the form $X\times$(Euclidean ball). We prove that if $X$ has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products $X\times$(Euclidean sphere). We give an example of a surface $X$, with Gaussian curvature negative somewhere, such that the product $X\times{\mathbb R}$ contains stable soap bubbles of arbitrarily large enclosed volume which do not even project surjectively onto the $X$ factor.
For each k > 0 we find an explicit function f_k such that the topology of S inside the ball B(p,r) is `bounded' by f_k(r) for every complete Riemannian surface (compact or noncompact) with K\geq -k^2, every point p on the surface, and every r. Using this result, we obtain a characterization (simple to check in practical cases) of the Gromov hyperbolicity of a Riemann surface S* (with its own Poincaré metric) obtained by deleting from one original surface S any uniformly separated union of continua and isolated points.