This paper investigates minimal n-dimensional submanifolds in the Euclidean space that are <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(n-2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-umbilic, meaning they carry an umbilical distribution of rank <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> arising from a nowhere vanishing principal normal vector field. We establish a correspondence between the class of minimal (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>)-umbilic submanifolds and the class of (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>)-singular minimal surfaces. These surfaces are the critical points of its "energy potential" and have been previously studied in various contexts, including physics and architecture where, for instance, they model surfaces with minimal potential energy under gravitational forces. We show that minimal, generic, (<InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>)-umbilic submanifolds, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, are (<InlineEquation ID="IEq10"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>)-rotational submanifolds whose profile is an (<InlineEquation ID="IEq11"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>)-singular minimal surface and vice versa. Furthermore, we develop a Weierstrass type method of local parametrization of all (<InlineEquation ID="IEq12"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>)-singular minimal surfaces, enabling a parametric description of all minimal n-dimensional, <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, hypersurfaces of the Euclidean space with a nowhere vanishing principal curvature of multiplicity <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Minimal $(n-2)$-Umbilic Submanifolds of the Euclidean Space

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发表日期：

2025年11月4日

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A. E. Kanellopoulou

摘要

This paper investigates minimal n-dimensional submanifolds in the Euclidean space that are $(n-2)$ $(n - 2)$ -umbilic, meaning they carry an umbilical distribution of rank $n-2$ $n - 2$ arising from a nowhere vanishing principal normal vector field. We establish a correspondence between the class of minimal ( $n-2$ $n - 2$ )-umbilic submanifolds and the class of ( $n-2$ $n - 2$ )-singular minimal surfaces. These surfaces are the critical points of its "energy potential" and have been previously studied in various contexts, including physics and architecture where, for instance, they model surfaces with minimal potential energy under gravitational forces. We show that minimal, generic, ( $n-2$ $n - 2$ )-umbilic submanifolds, $n\ge 4$ $n \geq 4$ , are ( $n-2$ $n - 2$ )-rotational submanifolds whose profile is an ( $n-2$ $n - 2$ )-singular minimal surface and vice versa. Furthermore, we develop a Weierstrass type method of local parametrization of all ( $n-2$ $n - 2$ )-singular minimal surfaces, enabling a parametric description of all minimal n-dimensional, $n\ge 4$ $n \geq 4$ , hypersurfaces of the Euclidean space with a nowhere vanishing principal curvature of multiplicity $n-2$ $n - 2$ .

Minimal \((n-2)\)-Umbilic Submanifolds of the Euclidean Space

摘要