This is a joint work with Franz Luef. The aim is to study the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander's dbar-L2 estimate with singular weight, Demailly's Calabi--Yau method for Kähler currents and a Kähler-variant of the symplectic embedding theorem of McDuff--Polterovich for ellipsoids. First, I will talk about the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. Then I will recall the Kähler geometry approach for the interpolation problem and explain our motivation for using ellipsoidal embedding. If the time is permitted, I will explain a sharp estimate of the Robin constant in the Ohsawa--Takegoshi approach for the one dimension interpolation bound.