<p>Given an open Riemann surface <i>M</i>, we prove that every nonflat conformal minimal immersion <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M\rightarrow \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>) is homotopic through nonflat conformal minimal immersions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$M\rightarrow \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> to a proper one. If <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M\rightarrow \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is homotopic to the real part of a proper holomorphic null embedding <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M\rightarrow \mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> directed by Oka cones in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>.</p>

Every Nonflat Conformal Minimal Surface is Homotopic to a Proper One

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2026年3月12日

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Tjaša Vrhovnik

摘要

Given an open Riemann surface M, we prove that every nonflat conformal minimal immersion $M\rightarrow \mathbb {R}^n$ $M \to R^{n}$ ( $n\ge 3$ $n \geq 3$ ) is homotopic through nonflat conformal minimal immersions $M\rightarrow \mathbb {R}^n$ $M \to R^{n}$ to a proper one. If $n\ge 5$ $n \geq 5$ , it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion $M\rightarrow \mathbb {R}^n$ $M \to R^{n}$ is homotopic to the real part of a proper holomorphic null embedding $M\rightarrow \mathbb {C}^n$ $M \to C^{n}$ . We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into $\mathbb {C}^n$ $C^{n}$ directed by Oka cones in $\mathbb {C}^n$ $C^{n}$ .