Quantized Einstein Metrics on $S^7$ with $SU(3)$-Symmetry

Abstract

We prove that the standard cohomogeneity-one action of \(SU(3)\) on \(S^7\), with principal orbit the Wallach flag manifold \(SU(3)/T^2\), admits infinitely many invariant Einstein metrics. The proof uses a detection approach to the Einstein boundary-value problem. Starting from one singular orbit, we follow the Einstein solutions only to a canonical hypersurface and measure there how far they are from closing smoothly. When the singular-orbit scale becomes small, this closing problem is governed by a Ricci-flat limiting solution. The linearisattion about its limiting cone has an oscillatory mode. As the scale shrinks, this oscillation repeatedly changes the sign of the closing error, producing infinitely many parameter values for which the metric closes. The closing scales satisfy an asymptotic logarithmic quantization law, with successive ratio tending to \(e^{-2π/\sqrt{15}}\). We also determine the geometry of the resulting sequence. Away from the two singular orbits the metrics converge to the singular sine cone over the non-normal Einstein metric on \(SU(3)/T^2\), while after rescaling by the square of the singular-orbit scale at either end they converge to the same complete Ricci-flat threshold metric. Their curvature is of order \(b_n^{-2}\), so the logarithmic phase law induces a corresponding quantization of the focal curvature scale.