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<p>Given a compact Lagrangian <i>L</i> in a semipositive convex-at-infinity symplectic manifold <i>W</i>, we establish a cup-length estimate for the action values of <i>L</i> associated to a Hamiltonian isotopy whose spectral norm is smaller than some <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3710_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(\hbar (L)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ħ</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. When <i>L</i> is rational, this implies a cup-length estimate on the number of intersection points. This Chekanov-type result generalizes a theorem of Kislev and Shelukhin proving non-displaceability in the case when <i>W</i> is closed and monotone. The method of proof is to deform the pair-of-pants product on Hamiltonian Floer cohomology using the Lagrangian <i>L</i>.</p>

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Lagrangian intersections and the spectral norm in convex-at-infinity symplectic manifolds

  • 期刊论文

  • 发表日期：

    2025年3月14日

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  • Habib Alizadeh,
  • Marcelo S. Atallah,
  • Dylan Cant

摘要

Given a compact Lagrangian L in a semipositive convex-at-infinity symplectic manifold W, we establish a cup-length estimate for the action values of L associated to a Hamiltonian isotopy whose spectral norm is smaller than some \(\hbar (L)\) ħ ( L ) . When L is rational, this implies a cup-length estimate on the number of intersection points. This Chekanov-type result generalizes a theorem of Kislev and Shelukhin proving non-displaceability in the case when W is closed and monotone. The method of proof is to deform the pair-of-pants product on Hamiltonian Floer cohomology using the Lagrangian L.