<p>If <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha \in \mathbb R$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stationary surface in Euclidean space is a surface <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> whose mean curvature <i>H</i> satisfies <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H(p)=\alpha |p|^{-2} \langle \nu ,p\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>α</mi> <mo stretchy="false">|</mo> <mi>p</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">⟨</mo> <mi>ν</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">⟩</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p\in \Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mi mathvariant="normal">Σ</mi> </mrow> </math></EquationSource> </InlineEquation>. These surfaces generalize in dimension two a classical family of curves studied by Euler which are critical points of the moment of inertia of planar curves. In this paper we establish, via inversions, a one-to-one correspondence between <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stationary surfaces and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$-(\alpha +4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-stationary surfaces. In particular, there is a correspondence between <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$-4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>-stationary surfaces and minimal surfaces. Using this duality we give some results of uniqueness of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$-4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>-stationary surfaces and we solve the Börling problem.</p>

A connection between minimal surfaces and the two-dimensional analogues of a problem of Euler

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2025年7月16日

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Rafael López

摘要

If $\alpha \in \mathbb R$ $α \in R$ , an $\alpha $ $α$ -stationary surface in Euclidean space is a surface $\Sigma $ $Σ$ whose mean curvature H satisfies $H(p)=\alpha |p|^{-2} \langle \nu ,p\rangle $ $H (p) = {α | p |}^{- 2} ⟨ ν, p ⟩$ , $p\in \Sigma $ $p \in Σ$ . These surfaces generalize in dimension two a classical family of curves studied by Euler which are critical points of the moment of inertia of planar curves. In this paper we establish, via inversions, a one-to-one correspondence between $\alpha $ $α$ -stationary surfaces and $-(\alpha +4)$ $- (α + 4)$ -stationary surfaces. In particular, there is a correspondence between $-4$ $- 4$ -stationary surfaces and minimal surfaces. Using this duality we give some results of uniqueness of $-4$ $- 4$ -stationary surfaces and we solve the Börling problem.