arXiv subjects
Publications and source records attributed to Pekka Pankka.
We construct, for a locally compact metric space $X$, a space of polylipschitz forms $\barΓ^*_c(X)$, which is a pre-dual for the space of metric currents of $\mathscr{D}_*(X)$ Ambrosio and Kirchheim. These polylipschitz forms may be seen as a substitute of differential forms in the metric setting.
We show that, for each $1\le p < 2$, there exists a wild involution $\mathbb S^3\to \mathbb S^3$ in the Sobolev class $W^{1,p}(\mathbb S^3,\mathbb S^3)$.
Let $f\colon M \to M$ be a uniformly quasiregular self-mapping of a compact, connected, and oriented Riemannian $n$-manifold $M$ without boundary, $n\ge 2$. We show that, for $k \in \{0,\ldots, n\}$, the induced homomorphism $f^* \colon H^k(M;\mathbb{R}) \to H^k(M;\mathbb{R})$, where $H^k(M;\mathbb{R})$ is the $k$:th singular cohomology of $M$, is complex diagonalizable and the eigenvalues of $f^*$ have modulus $(\mathrm{deg}\ f)^{k/n}$. As an application, we obtain a degree restriction for uniformly quasiregular self-mappings of closed manifolds. In the proof of the main theorem, we use a Sobolev--de Rham cohomology based on conformally invariant differential forms and an induced push-forward operator.
We give a version of Gromov's compactess theorem for pseudoholomorphic curves in the case of quasiregular mappings between closed manifolds. More precisely we show that, given $K\ge 1$ and $D\ge 1$, any sequence $(f_n \colon M \to N)$ of $K$-quasiregular mappings of degree $D$ between closed Riemannian $d$-manifolds has a subsequence which converges to a $K$-quasiregular mapping $f\colon X\to N$ of degree $D$ on a nodal $d$-manifold $X$.
Stoïlow's theorem from 1928 states that a continuous, open, and light map between surfaces is a discrete map with a discrete branch set. This result implies that such maps between orientable surfaces are locally modeled by power maps $z\mapsto z^k$ and admit a holomorphic factorization. The purpose of this expository article is to give a proof of this classical theorem having readers in mind that are interested in continuous, open and discrete maps.
Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping $f\colon X\to Y$ between oriented cohomology manifolds $X$ and $Y$ induces a pull-back operator $f^\ast \colon M_{k,loc}(Y) \to M_{k,loc}(X)$ between the spaces of metric $k$-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward $f_\ast \colon M_{k,loc}(X)\to M_{k,loc}(Y)$. As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology $n$-manifolds $X$ admitting a BLD-mapping $\mathbb{R}^n \to X$.
We prove the following dichotomy: if $n=2,3$ and $f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)$ is not homotopic to a constant map, then there is an open set $Ω\subset\mathbb{S}^{n+1}$ such that $\mathrm{rank}\, df=n$ on $Ω$ and $f(Ω)$ is dense in $\mathbb{S}^n$, while for any $n\geq 4$, there is a map $f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)$ that is not homotopic to a constant map and such that $\mathrm{rank}\, df<n$ everywhere. The result in the case $n\geq 4$ answers a question of Larry Guth.
We prove that every non-constant quasiregular selfmap of the $n$-sphere $\mathbb{S}^{n}$ admits a harmonic extension to the hyperbolic space $\mathbb{H}^{n+1}$ for $n\ge 2$.
We show that, for each $n\ge 3$, there exists a smooth Riemannian metric $g$ on a punctured sphere $\mathbb{S}^n\setminus \{x_0\}$ for which the associated length metric extends to a length metric $d$ of $\mathbb{S}^n$ with the following properties: the metric sphere $(\mathbb{S}^n,d)$ is Ahlfors $n$-regular and linearly locally contractible but there is no quasiconformal homeomorphism between $(\mathbb{S}^n,d)$ and the standard Euclidean sphere $\mathbb{S}^n$.
We show that, if the local dimension of the branch set of a discrete and open mapping $f\colon M\to N$ between $n$-manifolds is less than $(n-2)$ at a point $y$ of the image of the branch set $fB_f$, then the local monodromy of $f$ at $y$ is perfect. In particular, for generalized branched covers between $n$-manifolds the dimension of $fB_f$ is exactly $(n-2)$ at the points of abelian local monodromy. As an application, we show that a generalized branched covering $f\colon M \to N$ of local multiplicity at most three between $n$-manifolds is either a covering or $fB_f$ has local dimension $(n-2)$.
We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension $<1$ are free. On the other hand we construct for any $ε>0$ examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension $<1+ε$.
We show that for a closed $n$-manifold $N$ admitting a quasiregular mapping from the Euclidean $n$-space the following are equivalent: (1) order of growth of $π_1(N)$ is $n$, (2) $N$ is aspherical, and (3) $π_1(N)$ is virtually $\mathbb{Z}^n$ and torsion free.
We show that given $n\ge 3$, $q\ge 1$, and a finite set $\{y_1,\ldots, y_q\}$ in $\mathbb R^n$ there exists a quasiregular mapping $\mathbb R^n \to \mathbb R^n$ omitting exactly points $y_1,\ldots, y_q$.
We show that a closed, connected, oriented, Riemannian $n$-manifold, admitting a branched cover of bounded length distortion from $\mathbb R^n$, has a virtually Abelian fundamental group.
We study the existence of geometrically controlled branched covering maps from $\mathbb R^3$ to open $3$-manifolds or to decomposition spaces $\mathbb S^3/G$, and from $\mathbb S^3/G$ to $\mathbb S^3$.
Benjamini and Schramm introduced the notion of distributional limit of a sequence of graphs with uniformly bounded valence and studied such limits in the case that the involved graphs are planar. We investigate distributional limits of sequences of Riemannian manifolds with bounded curvature which satisfy certain condition of quasi-conformal nature. We then apply our results to somewhat improve Benjamini's and Schramm's original result on the recurrence of the simple random walk on limits of planar graphs. For instance, as an application give a proof of the fact that for graphs in an expander family, the genus of each graph is bounded from below by a linear function of the number of vertices.
We consider decomposition spaces $\R^3/G$ that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on $\R^3/G$ constructed via modular embeddings into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space $\R^3/G\times \R^m$ imposes quantitative topological constraints, in terms of the circulation and growth, to the defining sequences for $\R^3/G$. We give a necessary condition and a sufficient condition for the existence of parametrization. The necessary condition answers a question of Heinonen and Semmes on quasisymmetric parametrizability of spaces associated to the Bing double. The sufficient condition gives new examples of quasispheres in $\bS^4$.
We consider accumulation of periodic points in local uniformly quasiregular dynamics. Given a local uniformly quasiregular mapping $f$ with a countable and closed set of isolated essential singularities and their accumulation points on a closed Riemannian manifold, we show that points in the Julia set are accumulated by periodic points. If, in addition, the Fatou set is non-empty and connected, the accumulation is by periodic points in the Julia set itself. We also give sufficient conditions for the density of repelling periodic points.