For any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and any closed manifold <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathcal {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\pi _{n+k}({\mathcal {N}}) \ne \{0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>π</mi> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">N</mi> <mo stretchy="false">)</mo> </mrow> <mo>≠</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish the existence of nontrivial n-harmonic maps from <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathbb {S}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\mathcal {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, these maps naturally appear as bubbling limits of p-harmonic maps with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p > n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, obtained by min-max constructions in the limit <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$p \rightarrow n^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">→</mo> <msup> <mi>n</mi> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>.

Existence of nontrival n-harmonic maps via min-max methods

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2026年7月3日

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Dorian Martino,
Katarzyna Mazowiecka,
Armin Schikorra

摘要

For any $n \ge 3$ $n \geq 3$ and any closed manifold ${\mathcal {N}}$ $N$ with $\pi _{n+k}({\mathcal {N}}) \ne \{0\}$ $π_{n + k} (N) \neq {0}$ for some $k \ge 0$ $k \geq 0$ , we establish the existence of nontrivial n-harmonic maps from ${\mathbb {S}}^n$ $S^{n}$ into ${\mathcal {N}}$ $N$ . When $k\ge 1$ $k \geq 1$ , these maps naturally appear as bubbling limits of p-harmonic maps with $p > n$ $p > n$ , obtained by min-max constructions in the limit $p \rightarrow n^+$ $p \to n^{+}$ .