On BLD-mappings with small distortion
aa r X i v : . [ m a t h . M G ] F e b ON BLD-MAPPINGS WITH SMALL DISTORTION
AAPO KAURANEN, RAMI LUISTO, AND VILLE TENGVALL
Dedicated to Professor Pekka Koskela on his th birthday Abstract.
We show that every L -BLD-mapping in a domain of R n isa local homeomorphism if L < √ K I ( f ) <
2. These bounds aresharp as shown by a winding map. Introduction
Mappings with L -bounded length distortion (abbr. L -BLD mappings )were originally introduced by Martio and V¨ais¨al¨a in [MV88] as continuous,sense-preserving, discrete, and open mappings f : Ω → R n (Ω ⊂ R n domain with n ≥ L − ℓ ( α ) ≤ ℓ ( f ◦ α ) ≤ Lℓ ( α )(1)for every path α in Ω and for a fixed constant L ≥
1, where ℓ ( γ ) denotesthe length of a path γ . Notice that no constant map satisfies (1). Thesemappings form a superclass of local L -bi-Lipschitz mappings which havebeen studied by several authors, see e.g. [DP15, HMZ18, LP14, HS02] andthe references therein. If a mapping is an L -BLD-mapping for some L ≥ BLD-mapping .Unlike local bi-Lipschitz maps, BLD-mappings do not need to be localhomeomorphisms. Indeed, for instance the map( r, θ, z ) w (cid:0) √ − r, θ, √ − z (cid:1) ( z ∈ R n − )(2)in cylindrical coordinates of R n defines a noninjective √ Theorem 1.
Every L -BLD-mapping f : Ω → R n (Ω ⊂ R n domain with n ≥ with L < √ is a local homeomorphism. Date : February 13, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
BLD-mappings, branch set, quasiregular mappings, localhomeomorphism.A.K. acknowledges the support of Academy of Finland, grant number 322441.R.L. was partially supported by the Academy of Finland (grant 288501 ‘
Geometry ofsubRiemannian groups ’) and by the European Research Council (ERC Starting Grant713998 GeoMeG ‘
Geometry of Metric Groups ’).The research of V.T. was supported by the Academy of Finland, project number 308759.
Mappings of L -bounded length distortion form also a subclass of L n − -quasiregular mappings, see [MV88, Lemma 2.3]. We recall that a mapping f : Ω → R n (Ω ⊂ R n domain with n ≥ K -quasiregular if it belongs to Sobolev space W ,nloc (Ω , R n ) and sat-isfies the following distortion inequality | Df ( x ) | n ≤ KJ f ( x ) a.e.for a given constant K ≥
1. Here and what follows | A | stands for the operator norm of a given n × n matrix A and J f ( x ) = det Df ( x )denotes the Jacobian determinant of the differential matrix Df ( x ). To everyquasiregular mapping we associate the well-known inner and outer distor-tion functions defined as follows K I ( x, f ) = ( | D f ( x ) | n J f ( x ) n − , if J f ( x ) > , otherwiseand K O ( x, f ) = ( | Df ( x ) | n J f ( x ) , if J f ( x ) > , otherwise,measuring how the infinitesimal geometry of n -dimensional balls is distortedby the mapping. Here and what follows D ♯ f ( x ) = (cof Df ( x )) T , where cof stands for cofactor matrix. The corresponding inner and outerdilatations are defined by K I ( f ) := ess sup x ∈ Ω K I ( x, f ) , and K O ( f ) := ess sup x ∈ Ω K O ( x, f ) . For the theory and basic properties of quasiregular mappings we refer to thestandard monographies [IM01, Ric93, Res89, Vuo88].By the generalized Liouville theorem of Gehring [Geh62] and Reshetnyak[Res67] non-constant 1-quasiregular mappings are restrictions of M¨obiustransformations. On the other hand, for the winding map in (2) we have K I ( w ) = 2 and K O ( w ) = 2 n − . In [MRV71, Remark 4.7] it was conjectured that any quasiregular mappingwith K I ( f ) < ε ( n ) > f : Ω → R n (Ω ⊂ R n with n ≥
3) with K I ( f ) < ε ( n )is a local homeomorphism, see also [Gol71]. The best known quantitativebound for the number ε ( n ) > branch set B f = { x ∈ Ω : f is not a local homeomorphism at x } N BLD-MAPPINGS WITH SMALL DISTORTION 3 of the mapping is geometrically nice. In particular, this is the case whenthe branch set contains a rectifiable curve, see e.g. [Ric93, p. 76]. We givea short proof showing that the conjecture holds also for L -BLD mappings.See also [HK00] for other related injectivity results for BLD mappings. Theorem 2.
Let f : Ω → R n (Ω ⊂ R n domain with n ≥ be an L -BLD-mapping. Then K I ( f ) ≥ i ( x, f ) for every x ∈ Ω . (3) Especially, if K I ( f ) < then f is a local homeomorphism. Notice that the conclusion of Theorem 1 does not hold for quasiregularmappings in general. In [MRV71] there is an example of a quasiregularmapping f : Ω → R n for which sup { i ( x, f ) : x ∈ Ω } = ∞ . Also, noticethat Theorem 1 is valid in dimension two but the same conclusion does nothold for planar quasiregular mappings as one can see by considering theholomorphic function f : C → C , f ( z ) = z . This function is obviously not a BLD-mapping. We also point out that theproof of Theorem 2 is based on the modulus of continuity of quasiregularmappings and on the L -radiality of BLD-mappings. The BLD-property isused here only for the L -radiality.Finally, we recall that by Zorich’s theorem [Zor67] every locally injective,entire quasiregular mapping in dimension n ≥ z exp( z ). Notethat in the BLD-setting every entire, locally homeomorphic BLD-mappingin R n , n ≥
2, is a bi-Lipschitz homeomorphism onto R n . See eg. [MV88,Lemma 4.3]. If we combine this with our main results we obtain the follow-ing: Corollary 4.
Let f : R n → R n , n ≥ , be an entire BLD-mapping with L < √ or K I ( f ) < . Then f is an L -bi-Lipschitz map onto R n . Proof of Theorem 1
In the proof of Theorem 1 we use the L -radiality property of mappingswith L -bounded length distortion, see [Lui17]. For a space X we use thefollowing notation: H j ( X ) = “ j th homology group of X ” H j ( X ) = “ j th Alexander-Spanier co-homology group of X ”.For definition and basic properties of these homology and co-homologygroups, as well as the suspension mappings used in the following proof,we refer to [Hat02]. Necessary information on the topological degree theoryused in the proof can be found from [Ric93, Chapter I]. AAPO KAURANEN, RAMI LUISTO, AND VILLE TENGVALL
Proof of Theorem 1.
Let f : Ω → R n be an L -BLD-mapping with L < √ B f = { x ∈ Ω : f is not a local homeomorphism at x } of f is nonempty and fix a point x ∈ B f . Without loss of generality wemay assume that x = = f ( x ).We take first the blow-up of f by defining a sequence of mappings g j : j Ω → R n , g j ( x ) = jf ( x/j ) , where j Ω := { jx ∈ R n | x ∈ Ω } ( j = 1 , , . . . ) . We note that by [MV88, Theorem 4.7] (see also [Lui17, Section 4]) themappings g j are L -BLD-mappings and contain a subsequence converginguniformly to an L -BLD-mapping g : R n → R n such that ∈ B g , g ( ) = , and g − ( { g ( ) } ) = . Furthermore, since the L -BLD-mapping f is L -radial at , that is, thereexists r > x ∈ B (0 , r ) we have L − k x k ≤ k f ( x ) k ≤ L k x k . Clearly, for g j we have L − k x k ≤ k g j ( x ) k ≤ L k x k for all x ∈ B (0 , jr ) . This implies that the blow-up mapping g satisfies L − k x k ≤ k g ( x ) k ≤ L k x k for all x ∈ R n . In particular, we note that the composition of g | ∂B ( ,r ) andthe radial projection map p : R n \ B ( , r/L ) → ∂B ( , r/L ) , p ( x ) = ( r/L ) x k x k is L -Lipschitz. Thus, the mapping h : S n − → S n − , h ( x ) = ( L/r )( p ◦ g | ∂B ( ,r ) )( rx )is L -Lipschitz. Now by a classical dilatation result [Gro99, Proposition 2.9,p. 30] under the assumption L < √ h ∗ : H n − ( S n − ) → H n − ( S n − )equals either ± id or the constant map.Since for all r > g | ∂B ( ,r ) avoids the origin,we note that the restrictions are in fact mutually homotopic in R n \ { } and homotopic to h . In particular since S n − is a homotopy retract of R n \ { } , we see that g | R n \{ } is homotopic to h × id R with the identification R n \ { } ≃ S n − × R .Finally we note that as a BLD-mapping from R n to R n , the mapping g extends into a branched cover ˆ g : S n → S n , see e.g. [Lui16]. Furthermore, bythe above arguments this mapping ˆ g is homotopic to the suspension of themap h . Therefore, the induced homomorphismˆ g ∗ : H n ( S n ) → H n ( S n ) N BLD-MAPPINGS WITH SMALL DISTORTION 5 equals either ± id or the constant map. But now by the universal coefficienttheorem [Hat02, p.190] the induced homomorphismˆ g ∗ : H n ( S n ) → H n ( S n )also equals either ± id or a constant map. As the degree of a branchedcover S n → S n is always non-zero, this implies that ˆ g has degree ±
1. Thus,the mapping ˆ g : S n → S n is injective. This is a contradiction with theassumption ∈ B f and so the original claim holds. (cid:3) Proof of Theorem 2
In what follows, for a given continuous mapping f : Ω → R n (Ω ⊂ R n domain with n ≥ local (topological) index at a point x ∈ Ω by i ( x, f ) ∈ Z . Werecall that as a sense-preserving, continuous, discrete and open mappingevery BLD-mapping satisfies i ( x, f ) ≥ x ∈ Ω.Moreover, for a given point x ∈ Ω we have i ( x, f ) = 1 if and only if f isa local homeomorphism. Thus, in order to proof Theorem 2 it suffices toshow that under the assumptions of the theorem we have K I ( f ) ≥ i ( x, f ) forevery x ∈ Ω. For the properties of local index used here and what follows,see [Ric93, Chapter 1].
Proof of Theorem 2.
Fix a point x ∈ Ω and denote µ := (cid:0) i ( x , f ) /K I ( f ) (cid:1) n − . By combining [MV88, Corollary 2.13] and [Ric93, Theorem III.4.7, p.72] wesee that there exists a radius r > B ≥ L − k x − y k ≤ k f ( x ) − f ( y ) k ≤ B k x − y k µ (5)for every y ∈ B ( x , r ). By letting y → x we get from (5) that µ ≤ K I ( f ) ≥ i ( x, f ) for every x ∈ Ω,(6)and the claim follows. (cid:3)
Remarks . We end this note with the following remarks:(a) In the planar case Theorem 2 implies Theorem 1. Indeed, if f : Ω → R (Ω ⊂ R domain)is an L -BLD-mapping with L < √ K -quasiregular with K ≤ L n − < . Especially, in the planar case we have K I ( f ) = K O ( f ) < . Thus, by Theorem 2 the mapping f is a local homeomorphism. AAPO KAURANEN, RAMI LUISTO, AND VILLE TENGVALL (b) We do not know what is the optimal outer dilatation bound for thelocal injectivity of BLD-mappings. However, the map( r, θ, z ) ( r, θ, z ) ( z ∈ R n − )in cylindrical coordinates would suggest that every BLD-mappingwith K O ( f ) < n ≥ References [DP15] D. Drasin and P. Pankka. Sharpness of Rickman’s Picard theorem in all dimen-sions.
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