We construct calibrated submanifolds in Euclidean space invariant under the action of a Lie group G. We first demonstrate the method used in this paper by reproducing the results about special Lagrangians in Harvey-Lawson (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982). We then show explicitly that an associative submanifold in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^7$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>7</mn> </msup> </math></EquationSource> </InlineEquation> invariant under the action of a maximal torus <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {T}^2 \subset \textrm{G}_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> <mo>⊂</mo> <msub> <mtext>G</mtext> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> has to be a special Lagrangian submanifold in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {C}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. Similarly, we also show that a Cayley submanifold in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^8$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>8</mn> </msup> </math></EquationSource> </InlineEquation> invariant under the action of a maximal torus <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {T}^3 \subset \text {Spin}(7)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>3</mn> </msup> <mo>⊂</mo> <mtext>Spin</mtext> <mrow> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has to be a special Lagrangian submanifold in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {C}^4$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation>. We construct coassociative submanifolds in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {R}^7$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>7</mn> </msup> </math></EquationSource> </InlineEquation> invariant under the action of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{Sp}(1)\subset \mathbb {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Sp</mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi mathvariant="double-struck">H</mi> </mrow> </math></EquationSource> </InlineEquation> with a more general ansatz than the one in (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982) but we recover exactly the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textrm{Sp}(1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Sp</mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-invariant coassociatives in (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982), giving us a rigidity result. Finally, we construct cohomogeneity two examples of coassociative submanifolds in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {R}^7$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>7</mn> </msup> </math></EquationSource> </InlineEquation> which are invariant under the action of a maximal torus <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathbb {T}^2 \subset \textrm{G}_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> <mo>⊂</mo> <msub> <mtext>G</mtext> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>.

Some Results on Calibrated Submanifolds in Euclidean Space of Cohomogeneity One and Two

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发表日期：

2026年2月24日

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Faisal Romshoo

摘要

We construct calibrated submanifolds in Euclidean space invariant under the action of a Lie group G. We first demonstrate the method used in this paper by reproducing the results about special Lagrangians in Harvey-Lawson (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982). We then show explicitly that an associative submanifold in $\mathbb {R}^7$ $R^{7}$ invariant under the action of a maximal torus $\mathbb {T}^2 \subset \textrm{G}_2$ $T^{2} \subset G_{2}$ has to be a special Lagrangian submanifold in $\mathbb {C}^3$ $C^{3}$ . Similarly, we also show that a Cayley submanifold in $\mathbb {R}^8$ $R^{8}$ invariant under the action of a maximal torus $\mathbb {T}^3 \subset \text {Spin}(7)$ $T^{3} \subset Spin (7)$ has to be a special Lagrangian submanifold in $\mathbb {C}^4$ $C^{4}$ . We construct coassociative submanifolds in $\mathbb {R}^7$ $R^{7}$ invariant under the action of $\textrm{Sp}(1)\subset \mathbb {H}$ $Sp (1) \subset H$ with a more general ansatz than the one in (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982) but we recover exactly the $\textrm{Sp}(1)$ $Sp (1)$ -invariant coassociatives in (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982), giving us a rigidity result. Finally, we construct cohomogeneity two examples of coassociative submanifolds in $\mathbb {R}^7$ $R^{7}$ which are invariant under the action of a maximal torus $\mathbb {T}^2 \subset \textrm{G}_2$ $T^{2} \subset G_{2}$ .