For a sequence of approximate harmonic maps <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2072_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\{u_n\}_{n \in {\mathbb {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> from a compact Riemann surface M with boundary <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2072_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\partial M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> into a compact Riemannian manifold N with free boundaries lying on a closed supporting submanifold <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2072_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {K} \hookrightarrow N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">K</mi> <mo stretchy="false">↪</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> with uniformly bounded energy and with controlled tension fields, we study the lower and upper semi-continuity of Morse index and nullity for the sequence <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2072_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\{u_n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>.

Morse Index Semi-Continuity for Sequences of Approximate Harmonic Maps with Free Boundaries

期刊论文
发表日期：

2025年6月25日

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Rui Gao,
Miaomiao Zhu

摘要

For a sequence of approximate harmonic maps $\{u_n\}_{n \in {\mathbb {N}}}$ ${u_{n}}_{n \in N}$ from a compact Riemann surface M with boundary $\partial M$ $\partial M$ into a compact Riemannian manifold N with free boundaries lying on a closed supporting submanifold $\mathcal {K} \hookrightarrow N$ $K ↪ N$ with uniformly bounded energy and with controlled tension fields, we study the lower and upper semi-continuity of Morse index and nullity for the sequence $\{u_n\}$ ${u_{n}}$ .