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.

Variational convergence of discrete elasticae

We discuss a discretization of the Euler–Bernoulli bending energy and of Euler elasticae under clamped boundary conditions by polygonal lines. We show Hausdorff convergence of the set of almost minimizers of the discrete bending energy to the set of smooth Euler elasticae under mesh refinement in (i) the $W^{1,\infty }$-topology for piecewise-linear interpolation; and in (ii) the $W^{2,p}$-topology, $p \in [2,\infty [$, using a suitable smoothing operator to create $W^{2,p}$-curves from polygons.

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).

Influence of dimension on the convergence of level-sets in total variation regularization

We extend some recent results on the Hausdorff convergence of level-sets for total variation regularized linear inverse problems. Dimensions higher than two and measurements in Banach spaces are considered. We investigate the relation between the dimension and the assumed integrability of the solution that makes such an extension possible. We also give some counterexamples of practical application scenarios where the natural choice of fidelity term makes such a convergence fail.

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).