Topology of minimal surfaces in the sphere from capillarity

Abstract: We present a general construction of embedded minimal and constant mean curvature surfaces in $\mathbb{S}^n$ and one-phase free boundaries joined by a smooth interpolation by capillary hypersurfaces. This framework recovers all known families and produces new minimal surfaces in the sphere with rich topological structures as sphere bundles over base spaces which include space-form products, projective planes over division algebras, Stiefel manifolds, complex quadrics, and twisted products and quotients of Lie subgroups of $SO(n)$. We show these bundles are non-trivial and study their homotopy types using topological obstructions, including characteristic classes and tools from $K$-theory and stable homotopy theory. Finally, we prove uniqueness results for the rotationally invariant capillary CMC problem.