TY - JOUR

T1 - Fibers of Monotone Maps of Finite Distortion

AU - Kangasniemi, Ilmari

AU - Onninen, Jani

N1 - Funding Information:
J. Onninen was supported by the NSF Grant DMS-2154943.
Publisher Copyright:
© 2022, Mathematica Josephina, Inc.

PY - 2022/12

Y1 - 2022/12

N2 - We study topologically monotone surjective W1,n-maps of finite distortion f: Ω → Ω ′, where Ω , Ω ′ are domains in Rn, n≥ 2. If the outer distortion function Kf∈Llocp(Ω) with p≥ n- 1 , then any such map f is known to be homeomorphic, and hence the fibers f- 1{ y} are singletons. We show that as the exponent of integrability p of the distortion function Kf increases in the range 1 / (n- 1) ≤ p< n- 1 , then for increasingly many k∈ { 0 , ⋯ , n} depending on p, the k:th rational homology group Hk(f- 1{ y} ; Q) of any reasonably tame fiber f- 1{ y} of f is equal to that of a point. In particular, if p≥ (n- 2) / 2 then this is true for all k∈ { 0 , ⋯ , n}. We also formulate a Sobolev realization of a topological example by Bing of a monotone f: R3→ R3 with homologically non-trivial fibers. This example has Kf∈Lloc1/2-ε(R3) for all ε> 0 , which shows that our result is sharp in the case n= 3.

AB - We study topologically monotone surjective W1,n-maps of finite distortion f: Ω → Ω ′, where Ω , Ω ′ are domains in Rn, n≥ 2. If the outer distortion function Kf∈Llocp(Ω) with p≥ n- 1 , then any such map f is known to be homeomorphic, and hence the fibers f- 1{ y} are singletons. We show that as the exponent of integrability p of the distortion function Kf increases in the range 1 / (n- 1) ≤ p< n- 1 , then for increasingly many k∈ { 0 , ⋯ , n} depending on p, the k:th rational homology group Hk(f- 1{ y} ; Q) of any reasonably tame fiber f- 1{ y} of f is equal to that of a point. In particular, if p≥ (n- 2) / 2 then this is true for all k∈ { 0 , ⋯ , n}. We also formulate a Sobolev realization of a topological example by Bing of a monotone f: R3→ R3 with homologically non-trivial fibers. This example has Kf∈Lloc1/2-ε(R3) for all ε> 0 , which shows that our result is sharp in the case n= 3.

KW - Conformal cohomology

KW - Fiber

KW - Homology

KW - Mappings of finite distortion

KW - MFD

KW - Monotone

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U2 - 10.1007/s12220-022-01038-3

DO - 10.1007/s12220-022-01038-3

M3 - Article

AN - SCOPUS:85138753477

VL - 32

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 12

M1 - 299

ER -