A new approach for constructing minimal submanifolds of codimension 1 in the round spheres is proposed. In the case of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {S}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> two immersions of the Clifford torus and all Lawson <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\tau _{n, m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>τ</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> surfaces are described in terms of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\lambda , \mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-eigenfunctions. Also, a new proof of a theorem that describes <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(\lambda , \mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-eigenfunctions on the sphere is obtained. This proof is based on a statement that a function f is a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\lambda , \mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-eigenfunction if and only if f and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> are eigenfunctions for the Laplace-Beltrami operator.

Minimal submanifolds in spheres and complex-valued eigenfunctions

期刊论文
发表日期：

2026年7月22日

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Aleksei Kislitsyn

摘要

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