Sharpness of Rickman’s Picard theorem in all dimensions

Drasin, David; Pankka, Pekka

doi:10.1007/s11511-015-0125-x

Cited by 21 publications

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“…Rickman [20] proved the existence of a constant q = q(m, K) such that if a K-quasiregular mapping f : R m → R m omits at least q values in R m , then f is constant. This number q is called Rickman's constant, and this result becomes an extension of Picard's Theorem in the plane; for fixed m ≥ 3, [11] shows that q(m, K) → ∞ as K → ∞, the case m = 3 being due to Rickman [21]. Miniowitz obtained an analogue of Montel's Theorem for quasiregular mappings with poles, i.e.…”

Section: Quasiregular Maps a Continuous Mappingmentioning

confidence: 99%

The fast escaping set for quasiregular mappings

2014

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The fast escaping set of a transcendental entire function is the set of all points which tend to infinity under iteration as fast as compatible with the growth of the function. We study the analogous set for quasiregular mappings in higher dimensions and show, among other things, that various equivalent definitions of the fast escaping set for transcendental entire functions in the plane also coincide for quasiregular mappings. We also exhibit a class of quasiregular mappings for which the fast escaping set has the structure of a spider's web.Comment: 13 page

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Section: Quasiregular Maps a Continuous Mappingmentioning

confidence: 99%

The fast escaping set for quasiregular mappings

2014

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“…In higher dimensions, Rickman [18] proved what is now known as the Rickman-Picard theorem, showing that a K-quasiregular map from R d to the d-dimensional sphere S d can omit at most C(d, K) points. The fact that the constant depends on K is unavoidable as seen in the constructions by Rickman [19] and Drasin and Pankka [5].…”

Section: Eden Prywesmentioning

confidence: 99%

A bound on the cohomology of quasiregularly elliptic manifolds

Prywes

2019

Ann. of Math. (2)

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We show that a closed, connected and orientable Riemannian manifold of dimension d that admits a quasiregular mapping from R d must have bounded cohomological dimension independent of the distortion of the map. The dimension of the degree l de Rham cohomology of M is bounded above by d l . This is a sharp upper bound that proves the Bonk-Heinonen conjecture [2]. A corollary of this theorem answers an open problem posed by Gromov in 1981 [8]. He asked whether there exists a d-dimensional, simply connected manifold that does not admit a quasiregular map from R d . Our result gives an affirmative answer to this question. EDEN PRYWESThe sphere S 2 satisfies dim H 2 (S 2 ) = 1. By the Künneth formula [3, p. 47], dim H 2 (S 2 × S 2 ) = 2. For 1 ≤ l ≤ d − 2, H l (M #N ) ∼ = H l (M ) ⊕ H l (N ), whenever M and N are smooth manifolds by the Mayer-Vietoris Theorem [3, p. 22]. Therefore dim H 2 (M ) = 2n > 4 2 . So by Theorem 1.1, M is not quasiregularly elliptic.Theorem 1.1 is a generalization of a classical theorem for holomorphic functions in dimension 2. Let M be a Riemann surface, by the uniformization theorem, the universal cover of M is either C, C, or D. If f : C → M is holomorphic, then f lifts to a holomorphic map from C to the universal cover of M . If the universal covering space is D, then Liouville's theorem states that f is constant. This implies that the only compact Riemann surfaces that admit holomorphic mappings are homeomorphic to C and S 1 × S 1 . This proof can be applied to quasiregular mappings in dimension 2 because every quasiregular mapping f = g • φ, where g is holomorphic and φ : C → C is a quasiconformal homeomorphism [15, p. 247].A 1-quasiregular map on C is a holomorphic function. If we study quasiregular ellipticity for K = 1 in higher dimensions, then the results are as restrictive as in the d = 2 case. If M admits a 1-quasiregular mapping from R d , Bonk and Heinonen [2, Proposition 1.4] showed that M must be a quotient of the d-dimensional sphere or torus. For manifolds of dimension 3, Theorem 1.1 is known for each K ≥ 1. Jormakka [13] showed that if M is quasiregularly elliptic then M must be a quotient of S 3 , T 3 , or S 2 × S 1 . One sees that in higher dimensions there are separate results for when K = 1 and when K ≥ 1. In the study of K-quasiregular mappings for d ≥ 4, there are very few conditions on the topology of M that restrict which manifolds can be quasiregularly elliptic, independent of K.A theorem by Varopoulos gives a K-independent result. It states that the polynomial order of growth of the Cayley graph of the fundamental group of a quasiregularly elliptic manifold is bounded by d (see [21, Theorem X.5.1] or [9, Chapter 6]). This result gives a K-independent bound on the size of the fundamental group of the manifold, but does not apply when the fundamental group is small, specifically when the manifold is simply connected.A recent theorem due to Kangasniemi [14] gives a K-independent bound on the cohomology for manifolds that admit uniformly quasiregular self-mappings. H...

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“…Rickman also provided a converse in three dimensions [66]. It was only in a remarkable recent paper of Drasin and Pankka [21] that Rickman's construction was extended to all dimensions. Alternate PDE proofs of the Rickman-Picard Theorem were provided by Lewis and Eremenko-Lewis [52,23].…”

Section: Introductionmentioning

confidence: 97%

Heisenberg quasiregular ellipticity

2019

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Following the Euclidean results of Varopoulos and Pankka-Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold M to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group H. As an application, we show that a link complement S 3 \L has a sub-Riemannian metric admitting such a mapping only if L is empty, the unknot or Hopf link. In the converse direction, if L is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from H to S 3 \L.The main result is obtained by translating a growth condition on π 1 (M ) into the existence of a supersolution to the 4-harmonic equation, and relies on recent advances in the study of analysis and potential theory on metric spaces.

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Sharpness of Rickman’s Picard theorem in all dimensions

Abstract: We show that given n ≥ 3, q ≥ 1, and a finite set {y 1 , . . . , yq} in R n there exists a quasiregular mapping R n → R n omitting exactly points y 1 , . . . , yq.

Cited by 21 publications

References 20 publications

The fast escaping set for quasiregular mappings

The fast escaping set for quasiregular mappings

A bound on the cohomology of quasiregularly elliptic manifolds

Heisenberg quasiregular ellipticity

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