As a complement to Alain Valette’s article, this appendix gives an extension from group actions to groupoids of Proposition 2.4. Namely, it is shown that, for a smooth, second countable, locally compact groupoid endowed with a Borel Haar system, measurewise amenability is equivalent to amenability of the isotropy subgroups.
For any irreducible character χ of a finite group G , let θ ( χ ) denote the proportion of elements g ∈ G for which χ ( g ) is either zero or a root of unity. Then for any L ∈ [ 1 2 , 1 ] and any ϵ > 0 , there exists an irreducible character χ of a finite group such that | θ ( χ ) - L | < ϵ .
We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge–Ampère equation on either cylindrical type domains or the whole complex Euclidean space ℂ 2 . Among these, the entire solutions defined on ℂ 2 induce flat Kähler metrics, as expected by a question of Calabi. In contrast, those on cylindrical domains produce a family of nowhere flat Kähler metrics. Beyond these smooth solutions, we also classify solutions that are radially symmetric in one variable, which exhibit various types of singularities. Finally, we explore analogous solutions to Donaldson’s equation motivated by a result of He.
We show that the Monge-Ampère eigenfunctions of general bounded convex domains are globally Lipschitz. The same result holds for convex solutions to degenerate Monge-Ampère equations of the form D^2 u =M|u|^p with zero boundary condition on general bounded convex domains in ℝ^n within the sharp threshold p>n-2. As a consequence, we obtain global W^2, 1 estimates for these solutions.
