Conformal maps and p-Dirichlet energies

Let Φ: (M1, g1) → (M2, g2) be a diffeomorphism between Riemannian manifolds and Φ # : D(M2) → D(M1) the induced pull-back operator. The main theorem of this work is Theorem 4.1 which relates preservation of the p-Dirichlet energies ϕ 7 → ∫Mi |dϕ|p dµi under Φ # to isometric or conformal properties of Φ. More precisely: In case p = n, Φ # preserves the p-Dirichlet energy if and only if Φ is conformal. In case n 6 = p ≥ 2, energy preservation is equivalent to Φ being an isometry