We construct moduli spaces of G-equivariant harmonic maps between spheres, where G is a subgroup of the orthogonal group. For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$G=\text {SU}(m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mtext>SU</mtext> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\text {U}(m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>U</mtext> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\text {Sp}(m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Sp</mtext> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\text {Sp}(m)\times \text {U}(1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Sp</mtext> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mtext>U</mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\text {Sp}(m)\times \text {Sp}(1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Sp</mtext> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mtext>Sp</mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we obtain identity theorems: (i) The restriction to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S^5$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mn>5</mn> </msup> </math></EquationSource> </InlineEquation> of any <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\text {SU}(m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SU</mtext> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-equivariant harmonic map of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$S^{2m-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\text {SU}(3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SU</mtext> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-equivariant and harmonic. (ii) Any <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\text {SU}(3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SU</mtext> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-equivariant harmonic map of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$S^5$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mn>5</mn> </msup> </math></EquationSource> </InlineEquation> can be uniquely extended to an <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\text {SU}(m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SU</mtext> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-equivariant harmonic map of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$S^{2m-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mrow> <mn>2</mn> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. For the other groups, we have similar results.

Identity theorems for equivariant harmonic maps between spheres

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2026年1月19日

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Yasuyuki Nagatomo,
Isami Koga,
Masaro Takahashi

摘要

We construct moduli spaces of G-equivariant harmonic maps between spheres, where G is a subgroup of the orthogonal group. For $G=\text {SU}(m)$ $G = SU (m)$ , $\text {U}(m)$ $U (m)$ , $\text {Sp}(m)$ $Sp (m)$ , $\text {Sp}(m)\times \text {U}(1)$ $Sp (m) \times U (1)$ or $\text {Sp}(m)\times \text {Sp}(1)$ $Sp (m) \times Sp (1)$ , we obtain identity theorems: (i) The restriction to $S^5$ $S^{5}$ of any $\text {SU}(m)$ $SU (m)$ -equivariant harmonic map of $S^{2m-1}$ $S^{2 m - 1}$ is $\text {SU}(3)$ $SU (3)$ -equivariant and harmonic. (ii) Any $\text {SU}(3)$ $SU (3)$ -equivariant harmonic map of $S^5$ $S^{5}$ can be uniquely extended to an $\text {SU}(m)$ $SU (m)$ -equivariant harmonic map of $S^{2m-1}$ $S^{2 m - 1}$ . For the other groups, we have similar results.