Topology of minimal surfaces in the sphere from capillarity

TL;DR

This paper introduces a unified method for constructing embedded minimal and constant mean curvature surfaces in spheres, revealing new topologically rich examples and analyzing their homotopy types using advanced topological tools.

Contribution

It provides a general framework that recovers known surfaces and generates new examples with complex topology, employing topological obstructions and homotopy theory.

Findings

Constructed new minimal surfaces with rich topological structures in spheres.

Proved non-triviality of certain sphere bundles over base spaces.

Established uniqueness results for rotationally invariant capillary CMC surfaces.

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.