Metrics on a Surface with Bounded Total Curvature

Let $g=e^{2u}g_{euc}$ be a conformal metric defined on the unit disk of ${{\mathbb{C}}}$. We give an estimate of $\|\nabla u\|_{L^{2,\infty }(D_{\frac{1}{2}})}$ when $\|K(g)\|_{L^1}$ is small and $\frac{\mu (B_r^g(z),g)}{\pi r^2}<\Lambda $ for any $r$ and $z\in D_{\frac{3}{4}}$. Then we use this estimate to study the Gromov–Hausdorff convergence of a conformal metric sequence with bounded $\|K\|_{L^1}$ and give some applications.

Chordal Hausdorff Convergence and Quasihyperbolic Distance

We study Hausdorff convergence (and related topics) in the chordalization of a metric space to better understand pointed Gromov-Hausdorff convergence of quasihyperbolic distances (and other conformal distances).

On the Gromov-Hausdorff limit of metric spaces

In this paper, we show that a class of metric spaces determined by a continuous function f, which defines on the metric space of all real, n × n-matrices m is closed under the Gromov-Hausdorff convergence. This conclusion can be used to prove some metric properties of metric space is stable under the Gromov-Hausdorff convergence. Secondly, we consider the stability problem in Gromov hyperbolic space and show that if a sequence of Gromov hyperbolic spaces (Xn, dn) is said to converge to (X, d) in the sense of Gromov-Hausdorff convergence, then the Gromov hyperbolicity δ(Xn) of (Xn, dn) tends to the Gromov hyperbolicity δ(X) of (X, d).

Essential circles and Gromov–Hausdorff convergence of covers

The [Formula: see text]-covers of Sormani–Wei ([20]) are known not to be “closed” with respect to Gromov–Hausdorff convergence. In this paper we use the essential circles introduced in [19] to define a larger class of covering maps of compact geodesic spaces called “circle covers” that are “closed” with respect to Gromov–Hausdorff convergence and include [Formula: see text]-covers. In fact, we use circle covers to completely understand the limiting behavior of [Formula: see text]-covers. The proofs use the descrete homotopy methods developed by Berestovskii, Plaut, and Wilkins, and in fact we show that when [Formula: see text], the Sormani–Wei [Formula: see text]-cover is isometric to the Berestovskii–Plaut–Wilkins [Formula: see text]-cover. Of possible independent interest, our arguments involve showing that “almost isometries” between compact geodesic spaces result in explicitly controlled quasi-isometries between their [Formula: see text]-covers. Finally, we use essential circles to strengthen a theorem of E. Cartan by finding a new (even for compact Riemannian manifolds) finite set of generators of the fundamental group of a semilocally simply connected compact geodesic space. We conjecture that there is always a generating set of this sort having minimal cardinality among all generating sets.