Given a unit vector <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\vec {v}\in \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>v</mi> <mo stretchy="false">→</mo> </mover> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda ,\alpha \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>,</mo> <mi>α</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, a surface <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Sigma \subset \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Σ</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is called <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-singular minimal surface if its mean curvature H satisfies <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$H(p)=\alpha \frac{\langle N(p),\vec {v}\rangle }{\langle p,\vec {v}\rangle }+\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mfrac> <mrow> <mo stretchy="false">⟨</mo> <mi>N</mi> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mover accent="true"> <mi>v</mi> <mo stretchy="false">→</mo> </mover> <mo stretchy="false">⟩</mo> </mrow> <mrow> <mo stretchy="false">⟨</mo> <mi>p</mi> <mo>,</mo> <mover accent="true"> <mi>v</mi> <mo stretchy="false">→</mo> </mover> <mo stretchy="false">⟩</mo> </mrow> </mfrac> <mo>+</mo> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$p\in \Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mi mathvariant="normal">Σ</mi> </mrow> </math></EquationSource> </InlineEquation>, where N is the unit normal to <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>. In this paper, we study the axisymmetric solutions of this equation, identifying two types of surfaces depending on whether the rotation axis is parallel or orthogonal to <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\vec {v}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation>. For the first type, we prove the existence of surfaces that orthogonally intersect the rotation axis. For the second type, we provide a complete geometric description.

Axisymmetric solutions of the $\lambda $-singular minimal surface equation

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2026年6月26日

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Seher Kaya,
Rafael López

摘要

Given a unit vector $\vec {v}\in \mathbb {R}^3$ $\vec{v} \in R^{3}$ and $\lambda ,\alpha \in \mathbb {R}$ $λ, α \in R$ , a surface $\Sigma \subset \mathbb {R}^3$ $Σ \subset R^{3}$ is called $\lambda $ $λ$ -singular minimal surface if its mean curvature H satisfies $H(p)=\alpha \frac{\langle N(p),\vec {v}\rangle }{\langle p,\vec {v}\rangle }+\lambda $ $H (p) = α \frac{⟨ N (p), \vec{v} ⟩}{⟨ p, \vec{v} ⟩} + λ$ for all $p\in \Sigma $ $p \in Σ$ , where N is the unit normal to $\Sigma $ $Σ$ . In this paper, we study the axisymmetric solutions of this equation, identifying two types of surfaces depending on whether the rotation axis is parallel or orthogonal to $\vec {v}$ $\vec{v}$ . For the first type, we prove the existence of surfaces that orthogonally intersect the rotation axis. For the second type, we provide a complete geometric description.

Axisymmetric solutions of the \(\lambda \)-singular minimal surface equation

摘要