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On conjugate submersions

Antoni Pierzchalski
(Lodz)
Analysis Seminar
Thursday, 3 October, 2013, at 16:15, in Aud. D3 (1531-215)
Abstract:
The Newtonian capacity of a double connected domain in the planes equal to the conformal modulus of the family of curves connecting its boundary components. The conjugate family separates the components. The extremal functions of the families are harmonic and conjugate, so they define a holomorphic mapping. The level sets of the functions have a physical interpretation of being the force field lines and the equipotential lines, respectively. By the Dirichlet-Thomson principle the moduli of these two conjugate families are inverses (each to the other) or, in other words, their product is 1.

Inspired by this classical principle we define and investigate pairs of (p,q)--conjugate submersions of a Riemannian manifold, 1/p+1/q=1, and - in particular - of (p,q)--conjugate functions. We show that conjugate submersions of the plane are p-- and q--harmonic functions, respectively, and that - in the case on an arbitrary Riemannian manifold - the product of moduli of foliations defined by conjugate submersions is equal to 1.

The talk is based on a joint work with M. Ciska
Contact person: Bent Ørsted