Conformal Metrics on the Unit Ball in Euclidean Space

Abstract

We study densities ρ on the unit ball in euclidean space which satisfy a Harnack type inequality and a volume growth condition for the measure associated with ρ. For these densities a geometric theory can be developed which captures many features of the theory of quasiconformal mappings. For example, we prove generalizations of the Gehring‐Hayman theorem, the radial limit theorem and find analogues of compression and expansion phenomena on the boundary. 1991 Mathematics Subject Classification: 30C65.

Conformal metrics on the unit ball: The Gehring-Hayman property and the volume growth

“…Our final theorem is an extension of the Radial Limit Theorem [1,Theorem 4.4]. This result also relates to [3,Remark 1.3] where the size of the boundary set E ⊂ ∂B n where the conformal deformation mapping can "blow up" was estimated.…”

“…In [1, Theorem 3.1] it is namely shown that assuming a Harnack inequality and (1.1) guarantees that the geodesic arc in B n essentially is the shortest path with respect to ρ-distance between any x and y in B n . Nevertheless, the authors of [1] do not comment in any way whether (1.1) is the best possible upper bound for the volume growth or not.…”

“…However, not all conformal densities arise from a quasiconformal mapping; see [1] for more information and examples.…”

“…We also assume a growth condition on the isodiametric profile of (B n , d ρ ) which we, following [2], define as a function Notice that the condition η ρ (r) ≤ Br n for all r > 0 is equivalent to assuming the so-called volume growth condition [1], (1.1) µ ρ (B ρ (x, r)) ≤ Br n for all x ∈ B n and r > 0.…”

“…On the other hand, for arbitrary paths joining ξ and 0 in the unit ball, the situation is quite different. The only known relevant method, which has been used, e.g., in [1], is based on estimating the modulus of the families of paths and, as evaluation of examples like η ρ (r) = Br n (log(1 + 1/r)) p , p > 0, reveals, it does not allow us to relax exceedingly the volume growth condition for small r. However, if we replace (1.1) with the condition…”