Minimal equators and homological systoles in Berger projective spaces

Abstract

Gil-Medrano asked whether real projective subspaces remain minimal and volume-minimising under Berger deformations in dimensions greater than three. We answer the minimality question and make substantial progress on the corresponding minimisation problem. The fundamental observation is that the full linear geometry of a real subspace $V\subset\mathbb{C}^N$ is controlled by the single skew-adjoint endomorphism $A_V=\operatorname{pr}_VJ|_V$. For every non-round Berger metric, this yields a complete classification: $\mathbb{RP}(V)$ is minimal if and only if $V$ is a linear CR subspace, equivalently if all of its multiple K\"ahler angles are $0$ or $\pi/2$. Thus the answer to Gil-Medrano's minimality question is negative in general, although every equatorial hypersurface remains minimal. The same structure determines the critical points and global extrema of the linear volume functional. Combining it with projective integral geometry, we also compute exact mod-two homological systoles in the pure stretched and squashed regimes. Our final contribution is a conjecture that all remaining cases consist of mixed CR equators.