In this paper, we study surfaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$z=\varphi (x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in Euclidean space that satisfy the equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varphi _{xx}+\varphi _{yy}=\frac{\Lambda }{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>φ</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>φ</mi> <mrow> <mi mathvariant="italic">yy</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mi mathvariant="normal">Λ</mi> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Lambda \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is a real constant. We classify these surfaces when they are the zero level sets of an implicit equation of the type <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f(x)+g(y)+h(z)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where f, g and h are smooth functions of one variable. If <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Lambda =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we find a large family of surfaces with interesting symmetry properties. However, if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Lambda \not =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that the surfaces must be either surfaces of revolution or of the type <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$z=f(x)+g(y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>; furthermore, explicit parametrizations of these surfaces are obtained.

Separable surfaces that are critical points of the Dirichlet energy

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2026年3月23日

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Rafael López

摘要

In this paper, we study surfaces $z=\varphi (x,y)$ $z = φ (x, y)$ in Euclidean space that satisfy the equation $\varphi _{xx}+\varphi _{yy}=\frac{\Lambda }{2}$ $φ_{xx} + φ_{yy} = \frac{Λ}{2}$ where $\Lambda \in \mathbb {R}$ $Λ \in R$ is a real constant. We classify these surfaces when they are the zero level sets of an implicit equation of the type $f(x)+g(y)+h(z)=0$ $f (x) + g (y) + h (z) = 0$ , where f, g and h are smooth functions of one variable. If $\Lambda =0$ $Λ = 0$ , we find a large family of surfaces with interesting symmetry properties. However, if $\Lambda \not =0$ $Λ \neq 0$ , we show that the surfaces must be either surfaces of revolution or of the type $z=f(x)+g(y)$ $z = f (x) + g (y)$ ; furthermore, explicit parametrizations of these surfaces are obtained.