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FREQUENCY OF SOBOLEV AND QUASICONFORMAL DIMENSION DISTORTION
, 2010
"... We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev mapping f defined on a domain in R n, we estimate from above the Hausdorff dimension of the set of affine subspaces par ..."
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Cited by 9 (4 self)
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We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev mapping f defined on a domain in R n, we estimate from above the Hausdorff dimension of the set of affine subspaces parallel to a fixed mdimensional linear subspace, whose image under f has positive H α measure for some fixed α> m. As a consequence, we obtain new dimension distortion and absolute continuity statements valid for almost every affine subspace. Our results hold for mappings taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. Our theory extends to cover mappings in Sobolev–Lorentz spaces as well as pseudomonotone mappings in the critical Sobolev class. In particular, we obtain new absolute continuity statements for quasisymmetric maps from Euclidean domains into metric spaces. We illustrate our results with numerous examples.
SMOOTH QUASIREGULAR MAPS WITH BRANCHING IN R n
"... Abstract. According to a theorem of Rickman, all nonconstant C n/(n−2)smooth quasiregular maps in R n, n ≥ 3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in R 3. We prove that the order of smoothness is sharp in R 4. For each n ≥ 5 we construct a C 1+ɛ(n ..."
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Cited by 7 (1 self)
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Abstract. According to a theorem of Rickman, all nonconstant C n/(n−2)smooth quasiregular maps in R n, n ≥ 3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in R 3. We prove that the order of smoothness is sharp in R 4. For each n ≥ 5 we construct a C 1+ɛ(n)smooth quasiregular map in R n with nonempty branch set. 1.
Snowballs are Quasiballs
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2008
"... We introduce snowballs, which are compact sets in R³ homeomorphic to the unit ball. They are 3dimensional analogs of domains in the plane bounded by snowflake curves. For each snowball B a quasiconformal map f: R³ → R³ is constructed that maps B to the unit ball. ..."
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Cited by 5 (2 self)
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We introduce snowballs, which are compact sets in R³ homeomorphic to the unit ball. They are 3dimensional analogs of domains in the plane bounded by snowflake curves. For each snowball B a quasiconformal map f: R³ → R³ is constructed that maps B to the unit ball.
CONFORMAL Dimension does not . . .
"... We prove that the conformal dimension of any metric space is at least one unless it is zero. This confirms a conjecture of J. T. Tyson. ..."
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We prove that the conformal dimension of any metric space is at least one unless it is zero. This confirms a conjecture of J. T. Tyson.
QUASISYMMETRIC SPHERES OVER JORDAN DOMAINS
"... Abstract. Let Ω be a planar Jordan domain. We consider doubledomelike surfaces Σ defined by graphs of functions of dist(·, ∂Ω) over Ω. The goal is to find the right conditions on the geometry of the base Ω and the growth of the height so that Σ is a quasisphere, or quasisymmetric to S2. An interna ..."
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Abstract. Let Ω be a planar Jordan domain. We consider doubledomelike surfaces Σ defined by graphs of functions of dist(·, ∂Ω) over Ω. The goal is to find the right conditions on the geometry of the base Ω and the growth of the height so that Σ is a quasisphere, or quasisymmetric to S2. An internal uniform chordarc condition on the constant distance sets to ∂Ω, coupled with a mild growth condition on the height, gives a closetosharp answer. Our method also produces new examples of quasispheres in Rn, for any n ≥ 3. 1.