Related papers: A chord-arc covering theorem in Hilbert space

We say that a simple, closed curve $\gamma$ in the plane has bounded convex curvature if for every point $x$ on $\gamma$, there is an open unit disk $U_x$ and $\varepsilon_x>0$ such that $x\in\partial U_x$ and $B_{\varepsilon_x}(x)\cap…

Clearly, a generalized inverse limit of metrizable spaces indexed by $\mathbb N$ is metrizable, as it is a subspace of a countable product of metrizable spaces. The authors previously showed that all idempotent, upper semi-continuous,…

We prove the existence of a pair $(\Sigma ,\, \Gamma)$, where $\Sigma$ is a compact Riemann surface with $\text{genus}(\Sigma)\, \geq\, 2$, and $\Gamma\, \subset\, {\mathrm SL}(2, \mathbb C)$ is a cocompact lattice, such that there is a…

Consider a compact Lie group $G$ and a closed Lie subgroup $H<G$. Let $\mathcal M$ be the set of $G$-invariant Riemannian metrics on the homogeneous space $M=G/H$. By studying variational properties of the scalar curvature functional on…

Koberda proved that if a graph $\Gamma$ is a full subgraph of a curve graph $\mathcal{C}(S)$ of an orientable surface $S$, then the right-angled Artin group $A(\Gamma)$ on $\Gamma$ is a subgroup of the mapping class group ${\rm Mod}(S)$ of…

We show that if the curvature of a Cartan-Hadamard $n$-manifold is constant near a convex hypersurface $\Gamma$, then the total Gauss-Kronecker curvature $\mathcal{G}(\Gamma)$ is not less than that of any convex hypersurface nested inside…

Let $\Sigma$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $\Sigma$. A geodesic curve $\gamma$ in $\Sigma$ is filling, if it cuts the surface into topological disks and…

In this paper we prove a theorem concerning lamination limits of sequences of compact disks $M_n$ embedded in $\mathbb{R}^3$ with constant mean curvature $H_n$, when the boundaries of these disks tend to infinity. This theorem generalizes…

We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total…

If $U$ is a $C^{\infty}$ function with compact support in the plane, we let $u$ be its restriction to the unit circle $\mathbb{S}$, and denote by $U_i,\,U_e$ the harmonic extensions of $u$ respectively in the interior and the exterior of…

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em with no critical points}. This kind of result can be…

We show that, for a constant-degree algebraic curve $\gamma$ in $\mathbb{R}^D$, every set of $n$ points on $\gamma$ spans at least $\Omega(n^{4/3})$ distinct distances, unless $\gamma$ is an {\it algebraic helix} (see Definition 1.1). This…

We prove the three embeddedness results as follows. $({\rm i})$ Let $\Gamma_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of…

Suppose that $M$ is a $2$-dimensional oriented Riemannian manifold, and let $\gamma$ be a simple closed curve on $M$. Let $m \gamma$ denote the curve formed by tracing $\gamma$ $m$ times. We prove that if $m \gamma$ is contractible through…

The aim of the paper is to prove that if $M$ is a metrizable manifold modelled on a Hilbert space of dimension $\alpha \geq \aleph_0$ and $F$ is its $\sigma$-$Z$-set, then for every completely metrizable space $X$ of weight no greater than…

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of compact manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then $Y$ has a universal cover. We then show that, for $i$…

Let $N$ be a closed nonorientable surface with or without marked points. In this paper we prove that, for every finite full subgraph $\Gamma$ of $\mathcal{C}^{\mathrm{two}}(N)$, the right-angled Artin group on $\Gamma$ can be embedded in…

The equivariant coarse Novikov conjectures stand among a handful profound $K$-theoretic conjectures in noncommutative geometry. Motivated by the quest to verify Novikov-type conjectures for groups of diffeomorphisms, we study in this paper…

Let $M$ be an oriented even-dimensional Riemannian manifold on which a discrete group $\Gamma$ of orientation-preserving isometries acts freely, so that the quotient $X=M/\Gamma$ is compact. We prove a vanishing theorem for a half-kernel of…

Let $\Gamma$ be a finitely generated group which is hyperbolic relative to a finite family $\{H_1,...,H_n\}$ of subgroups. We prove that $\Gamma$ is uniformly embeddable in a Hilbert space if and only if each subgroup $H_i$ is uniformly…