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Minimal submanifolds in spheres and complex-valued eigenfunctions

Aleksei Kislitsyn

TL;DR

The paper links minimal submanifolds of spheres to complex-valued $(\lambda,\mu)$-eigenfunctions, proposing a construction of codimension-1 minimal submanifolds in $\mathbb{S}^n$ via level sets associated with real lines under a complex eigenfunction. It proves a concise classification (Theorem $\text{['Sn_class']}$) that any $(\lambda,\mu)$-eigenfunction on $\mathbb{S}^n$ comes from a homogeneous $(0,0)$-eigenfunction, and demonstrates how this yields explicit Clifford torus and Lawson $\tau_{n,m}$-surface immersions on $\mathbb{S}^3$. The work analyzes $\mathbb{S}^4$ examples showing many low-degree polynomial cases yield trivial minimal submanifolds, while outlining a general Hessian-based criterion for minimality of $f^{-1}(l_\alpha)$ and its consequences in $\mathbb{S}^3$. Overall, the paper strengthens the connection between spectral geometry and minimal submanifold theory on spheres and provides concrete constructions via $f^{-1}(l_\alpha)$ and polynomial $(0,0)$-eigenfunctions.

Abstract

A new approach for constructing minimal submanifolds of codimension 1 in the round spheres is proposed. In the case of $\mathbb{S}^3$ two immersions of the Clifford torus and all Lawson $τ_{n, m}$ surfaces are described in terms of $(λ, μ)$-eigenfunctions. Also, a new proof of a theorem that describes $(λ, μ)$-eigenfunctions on sphere is obtained. This proof is based on a statement that a function $f$ is a $(λ, μ)$-eigenfunction if and only if $f$ and $f^2$ are eigenfunctions for the Laplace-Beltrami operator.

Minimal submanifolds in spheres and complex-valued eigenfunctions

TL;DR

The paper links minimal submanifolds of spheres to complex-valued -eigenfunctions, proposing a construction of codimension-1 minimal submanifolds in via level sets associated with real lines under a complex eigenfunction. It proves a concise classification (Theorem ) that any -eigenfunction on comes from a homogeneous -eigenfunction, and demonstrates how this yields explicit Clifford torus and Lawson -surface immersions on . The work analyzes examples showing many low-degree polynomial cases yield trivial minimal submanifolds, while outlining a general Hessian-based criterion for minimality of and its consequences in . Overall, the paper strengthens the connection between spectral geometry and minimal submanifold theory on spheres and provides concrete constructions via and polynomial -eigenfunctions.

Abstract

A new approach for constructing minimal submanifolds of codimension 1 in the round spheres is proposed. In the case of two immersions of the Clifford torus and all Lawson surfaces are described in terms of -eigenfunctions. Also, a new proof of a theorem that describes -eigenfunctions on sphere is obtained. This proof is based on a statement that a function is a -eigenfunction if and only if and are eigenfunctions for the Laplace-Beltrami operator.
Paper Structure (9 sections, 7 theorems, 25 equations)

This paper contains 9 sections, 7 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Complex-valued eigenfunctions on sphere
  3. Auxiliary proposition
  4. Proof of Theorem \ref{['Sn_class']}
  5. Application to S4
  6. Minimal submanifolds of codimension 1 and complex-valued eigenfunctions
  7. Proof of Theorem \ref{['Sn_th']}
  8. Proof of Theorem \ref{['S3_th']}
  9. Examples in $\mathbb{S}^3$

Key Result

Theorem 1.2

Let $f: (M, g) \rightarrow \mathbb{C}$ be a complex-valued eigenfunction on a Riemannian manifold, such that $0 \in f(M)$ is a regular value for $f$. Then the fiber $f^{-1}(0)$ is a minimal submanifold of $M$ of codimension two.

Theorems & Definitions (19)

  • Definition 1.1: gudmundsson2023minimal
  • Theorem 1.2: gudmundsson2023minimal
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • proof
  • Example 3.1
  • Definition 3.2: gudmundsson2023minimal
  • ...and 9 more