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Topology of minimal surfaces in the sphere from capillarity

Benjy Firester, Raphael Tsiamis

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.

Topology of minimal surfaces in the sphere from capillarity

Abstract

We present a general construction of embedded minimal and constant mean curvature surfaces in 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 . We show these bundles are non-trivial and study their homotopy types using topological obstructions, including characteristic classes and tools from -theory and stable homotopy theory. Finally, we prove uniqueness results for the rotationally invariant capillary CMC problem.

Paper Structure

This paper contains 15 sections, 41 theorems, 235 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isoparametric hypersurfaces of the sphere
  4. The minimal surface equation
  5. Topology of the minimal surfaces
  6. Type I
  7. Type II
  8. Construction of capillary minimal surfaces
  9. ODE analysis
  10. Auxiliary quantities
  11. Type I
  12. Type II
  13. CMC hypersurfaces in the sphere
  14. The CMC equation
  15. Uniqueness of the axisymmetric capillary cone

Key Result

Theorem 1.1

For any isoparametric hypersurface $M \subset \mathbb{S}^{n-1}$ with at least two distinct principal curvatures, let $M_1, M_2$ be the associated focal manifolds. There exist three closed embedded minimal surfaces $\mathbf{S}_{M_1}, \mathbf{S}_{M_2}, \bar{\mathbf{S}}_M \subset \mathbb{S}^n$ of two c $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: We display numerically computed profile curves for varying angle Type I and II cones for $(g,m_1, m_2) = (4,2,5)$ corresponding to $\Sigma_{M_1, \theta}, \bar{\Sigma}_{M,\theta}, \Sigma_{M_2,\theta}$ respectively. The dashed lines indicate non-geometric Type I solutions to \ref{['eqn:ode-star']} which blow up before attaining 0. The smoothness condition for $\Sigma_{M_2,\theta}$ requires the compatibility condition $f_{M_2,\theta}'(1) = \frac{4(n-1)}{g^2(m_2+1)}f_{M_2,\theta}(1) > 0$ on the terminal data, which can be observed visually.
  • Figure 2: We display numerically computed profile curves for varying mean curvature $H$ for both Type I and II cones for $(g,m_1, m_2) = (4,2,5)$ corresponding to $\Sigma^H_{M_1, \frac{\pi}{2}}$. We can see some intervals of $H$ where the surfaces do not intersect, thereby laminating a region. However, this property fails for the minimal leaf $\Sigma_{M_1, \frac{\pi}{2}}$, which intersects the CMC profiles.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1
  • Definition 1
  • Proposition 2.1
  • proof
  • ...and 74 more