<p>This paper classifies invariant surfaces in the 3-dimensional solvable Lie group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\text {Sol}_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>Sol</mtext> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation> that act as solitons under the Gauss curvature flow. Specifically, we consider solitons associated with the canonical basis of Killing vector fields <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{F_1, F_2, F_3\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>F</mi> <mn>3</mn> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$F_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> generate horizontal translations and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$F_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation> generates a scaling isometry. Rigidity results are established for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$F_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-invariant surfaces, proving that certain totally geodesic vertical planes are the only <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$F_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$F_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-solitons. Finally, for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$F_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-invariant surfaces, we describe the main geometric properties of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$F_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$F_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-solitons, considering both extrinsic and intrinsic Gauss curvatures.</p>

Gauss Curvature Solitons on Invariant Surfaces in the Homogeneous Space Sol

期刊论文
发表日期：

2026年7月18日

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Rafael Belli,
Rafael López

摘要

This paper classifies invariant surfaces in the 3-dimensional solvable Lie group $\text {Sol}_3$ ${Sol}_{3}$ that act as solitons under the Gauss curvature flow. Specifically, we consider solitons associated with the canonical basis of Killing vector fields $\{F_1, F_2, F_3\}$ ${F_{1}, F_{2}, F_{3}}$ , where $F_1$ $F_{1}$ and $F_2$ $F_{2}$ generate horizontal translations and $F_3$ $F_{3}$ generates a scaling isometry. Rigidity results are established for $F_3$ $F_{3}$ -invariant surfaces, proving that certain totally geodesic vertical planes are the only $F_1$ $F_{1}$ - and $F_2$ $F_{2}$ -solitons. Finally, for $F_1$ $F_{1}$ -invariant surfaces, we describe the main geometric properties of $F_2$ $F_{2}$ - and $F_3$ $F_{3}$ -solitons, considering both extrinsic and intrinsic Gauss curvatures.