arXiv subjects

Ilmari Kangasniemi

Publications and source records attributed to Ilmari Kangasniemi.

23 records · Page 2Linked to original sources

arXiv2020

Conformally formal manifolds and the uniformly quasiregular non-ellipticity of $(\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$

We show that the manifold $(\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$ does not admit a non-constant non-injective uniformly quasiregular self-map. This answers a question of Martin, Mayer, and Peltonen, and provides the first example of a quasiregularly elliptic manifold which is not uniformly quasiregularly elliptic. To obtain the result, we introduce conformally formal manifolds, which are closed smooth $n$-manifolds $M$ admitting a measurable conformal structure $[g]$ for which the $(n/k)$-harmonic $k$-forms of the structure $[g]$ form an algebra. This is a conformal counterpart to the existing study of geometrically formal manifolds. We show that, similarly as in the geometrically formal theory, the real cohomology ring $H^*(M; \mathbb{R})$ of a conformally formal $n$-manifold $M$ admits an embedding of algebras $Φ\colon H^*(M; \mathbb{R}) \hookrightarrow \wedge^* \mathbb{R}^n$. We also show that uniformly quasiregularly elliptic manifolds $M$ are conformally formal in a stronger sense, in which the wedge product is replaced with a conformally scaled Clifford product. For this stronger version of conformal formality, the image of $Φ$ is closed under the Euclidean Clifford product of $\wedge^* \mathbb{R}^n$, which in turn is impossible for $M = (\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$.

math.CV↗

arXiv2020

Entropy in uniformly quasiregular dynamics

Let $M$ be a closed, oriented, and connected Riemannian $n$-manifold, for $n\ge 2$, which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map $f\colon M\to M$, the topological entropy $h(f)$ is $\log \mathrm{deg}( f )$. This proves Shub's entropy conjecture in this case.

math.DS↗

arXiv2019

Obstructions for automorphic quasiregular maps and Lattès-type uniformly quasiregular maps

Suppose that $M$ is a closed, connected, and oriented Riemannian $n$-manifold, $f \colon \mathbb{R}^n \to M$ is a quasiregular map automorphic under a discrete group $Γ$ of Euclidean isometries, and $f$ has finite multiplicity in a fundamental cell of $Γ$. We show that if $Γ$ has a sufficiently large translation subgroup $Γ_T$, then $\dim Γ\in \{0, n-1, n\}$. If $f$ is strongly automorphic and induces a non-injective Lattès-type uniformly quasiregular map, then the same holds without the assumption on the size of $Γ_T$. Moreover, an even stronger restriction holds in the Lattès case if $M$ is not a rational cohomology sphere.

math.DG↗

arXiv2019

Uniform cohomological expansion of uniformly quasiregular mappings

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.

math.CV↗

arXiv2017

Sharp cohomological bound for uniformly quasiregularly elliptic manifolds

We show that if a compact, connected, and oriented $n$-manifold $M$ without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring $H^*(M; \mathbb{R})$ of $M$ is bounded from above by $2^n$. This is a positive answer to a dynamical counterpart of the Bonk-Heinonen conjecture on the cohomology bound for quasiregularly elliptic manifolds. The proof is based on an intermediary result that, if $M$ is not a rational homology sphere, then each such uniformly quasiregular self-map on $M$ has a Julia set of positive Lebesgue measure.

math.CV↗