Half-volume Spectrum and Constant Mean Curvature Hypersurfaces

We will introduce the half-volume spectrum, a new variational invariant inspired by the classical volume spectrum, but defined by considering only hypersurfaces that divide the ambient manifold into equal volumes. We will present a new min-max theory for constant mean curvature (CMC) hypersurfaces specifically associated with the half-volume spectrum. As an application, we establish a CMC analogue of Yau’s abundance conjecture and show that it holds generically in low dimensions. This is based on joint work with Liam Mazurowski.