<p>In the joint work of the author with Da Lio, F., and Rivière, T., Schlagenhauf, D.: (2025) we studied the stability of the Morse index for Sacks-Uhlenbeck sequences into spheres as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p\searrow 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>↘</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. These are critical points of the energy <Equation ID="Equ86"> <EquationSource Format="TEX">$E_p(u) {:}{=}\int _\Sigma \left( 1+\left| \nabla u\right| ^2\right) ^{p/2} \ dvol_\Sigma ,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>E</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <msub> <mo>∫</mo> <mi mathvariant="normal">Σ</mi> </msub> <msup> <mfenced close=")" open="("> <mn>1</mn> <mo>+</mo> <msup> <mfenced close="|" open="|"> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mfenced> <mn>2</mn> </msup> </mfenced> <mrow> <mi>p</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mspace width="4pt" /> <mi>d</mi> <mi>v</mi> <mi>o</mi> <msub> <mi>l</mi> <mi mathvariant="normal">Σ</mi> </msub> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u:\Sigma \rightarrow S^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>:</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is a map from a closed Riemannian surface <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> into a sphere <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ S^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. In this paper we extend the results found in Da Lio, F., Rivière, T., Schlagenhauf, D.: (2025) to the case of Sacks-Uhlenbeck sequences into homogeneous spaces, by incorporating the strategy introduced in Bayer, C., and Roberts, A.: (2025) . In the spirit of Da Lio, F., Rivière, T., Schlagenhauf, D.: (2025), we show in this setting the upper semicontinuity of the Morse index plus nullity and an improved pointwise estimate of the gradient in the neck regions around blow up points.</p>

Stability of the morse index for the p-harmonic approximation of harmonic maps into homogeneous spaces

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2026年1月20日

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Dominik Schlagenhauf

摘要

In the joint work of the author with Da Lio, F., and Rivière, T., Schlagenhauf, D.: (2025) we studied the stability of the Morse index for Sacks-Uhlenbeck sequences into spheres as $p\searrow 2$ $p ↘ 2$ . These are critical points of the energy $E_p(u) {:}{=}\int _\Sigma \left( 1+\left| \nabla u\right| ^2\right) ^{p/2} \ dvol_\Sigma ,$ $E_{p} (u) : = \int_{Σ} {(1 + {|\nabla, u|}^{2})}^{p / 2} d v o l_{Σ},$ where $u:\Sigma \rightarrow S^n$ $u : Σ \to S^{n}$ is a map from a closed Riemannian surface $\Sigma $ $Σ$ into a sphere $ S^n$ $S^{n}$ . In this paper we extend the results found in Da Lio, F., Rivière, T., Schlagenhauf, D.: (2025) to the case of Sacks-Uhlenbeck sequences into homogeneous spaces, by incorporating the strategy introduced in Bayer, C., and Roberts, A.: (2025) . In the spirit of Da Lio, F., Rivière, T., Schlagenhauf, D.: (2025), we show in this setting the upper semicontinuity of the Morse index plus nullity and an improved pointwise estimate of the gradient in the neck regions around blow up points.