Modulus of revolution rings in the heisenberg group

Abstract

Abstract. Let S be a surface of revolution embedded in the Heisenberg group H. A revolution ring R a,b (S), 0 < a < b, is a domain in H bounded by two dilated images of S, with dilation factors a and b, respectively. We prove that if S is subject to certain geometric conditions, then the modulus of the family Γ of horizontal boundary connecting curves inside R a,b (S) isOur result applies for many interesting surfaces, e.g., the Korányi metric sphere, the CarnotCarathéodory metric sphere and the bubble set.

Quasiconformal mappings on the Heisenberg group: An overview

“…But in the Heisenberg group case things are rather different: In [27], Korányi and Reimann calculated the capacity (that is, the modulus of the family of curves joining the two components of the boundary) of a ring whose boundary comprises two homocentric Korányi-Cygan spheres. Rather surprisingly, a variety of rings centered at the origin (like rings whose boundary comprises of two Carnot-Carathéodory spheres), have exactly the same capacity, see [33]. In general though, the calculation of moduli of curve families in an arbitrary ring inside H is a difficult task.…”

The modulus of the Korányi ellipsoidal ring

“…A Korányi spherical ring is a domain bounded between two d H 1 metric spheres of radii B and A, 0 < B < A. Korányi and Reimann proved in [3] that its modulus is equal to π 2 (log(A/B)) −3 . Platis also showed in [7] that the modulus of a revolution ring, that is, a domain bounded between two dilated images D B (S) and D A (S) of a surface of revolution S, is also equal to π 2 (log(A/B)) −3 .…”