Volume-Minimising Projective Subspaces in Berger Projective Spaces: Three Further Cases of a Question of Gil-Medrano

Gil-Medrano asked whether, in the real projective space \(\mathbb{RP}^{2n+1}\) with a Berger metric \(g_\mu\) (the round metric with the Hopf fibres rescaled by \(\sqrt\mu\)), the projective subspaces coming from equatorial \(k\)-spheres are minimal, and whether they are the only volume-minimising \(k\)-cycles in their homology class. A recent preprint of G. Wheeler answers the first question, determines the volume minimisers among projective subspaces, proves that they minimise among all cycles in two regimes, and conjectures this in the remaining ones. This note is a partial answer to the second question: following a route proposed in that preprint, Crofton formulas adapted to the unitary group, we prove three further cases of the conjecture, for the classes of competitors named below. (I) For every \(n\) and every \(\mu>0\) the projective hyperplanes minimise volume; the proof is an exact Crofton formula with great circles and the positive weight \((1+(\mu-1)\kappa^2)^{-(n+1)}\), where \(\kappa\) is the cosine of the Kähler angle of the circle. (II) For \(0<\mu<1\) and every even dimension \(k=2j\le 2n\) the subspaces of type \(\mathbb{C}^j\oplus\mathbb{R}\) minimise; the slices are common kernels of \(j\) pairs of Gaussian functionals correlated through the complex structure, mixed by an explicit positive measure. (III) In \(\mathbb{RP}^5\), for every \(\mu>1\), the \(3\)-dimensional subspaces of type \(\mathbb{C}\oplus\mathbb{R}^2\) minimise; the measure is found by continuing the Gaussian model analytically to negative parameters. In each case we give the value of the minimum, prove the inequality for countably rectifiable sets that meet almost every complementary projective subspace (in particular for compact embedded \(C^1\) submanifolds in the non-zero class mod 2 and for Lipschitz cycles mod 2), and prove uniqueness among compact embedded \(C^1\) submanifolds. Together with Wheeler's theorems this determines the least volume and the minimisers, in these classes, for \(0<\mu<1\) in all dimensions and for all \(\mu\) in \(\mathbb{RP}^5\). The case \(\mu>1\), \(n<k<2n\), \(n\ge 3\) remains open. This is an unrefereed note.