Remarks on variational problems for Fefferman's measure

Abstract

We investigate the Plateau and isoperimetric problems associated to Fefferman's measure for strongly pseudoconvex real hypersurfaces in C n (focusing on the case n = 2), showing in particular that the isoperimetric problem shares features of both the euclidean isoperimetric problem and the corresponding problem in Blaschke's equiaffine geometry in which the key inequalities are reversed.The problems are invariant under constant-Jacobian biholomorphism, but we also introduce a non-trivial modified isoperimetric quantity invariant under general biholomorphism.

Volume Approximations of Strongly Pseudoconvex Domains

“…where F is a biholomorphism on Ω that is C 2 -smooth on Ω. The Fefferman hypersurface measure shares strong connections with the Blaschke surface area measure (explored in [3] and [4], for instance) studied in affine convex geometry. If K ⊂ R d is a C 2 -smooth convex body, the Blaschke surface area measure on ∂K is given by σK = κ 1 d+1 s Euc , where κ and s Euc are the Gaussian curvature function and the Euclidean surface area form on ∂K, respectively.…”

“…• We will use z (and similarly w) to denote both (z 1 , z 2 ) = (x 1 +iy 1 , x 2 +iy 2 ) ∈ C 2 and (x 1 , y 1 , x 2 , y 2 ) ∈ R 4 . The usage will be clear from the context.…”