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<p>On a two-dimensional Riemannian manifold without boundary we consider the variational limit of a family of functionals given by the sum of two terms: a Ginzburg–Landau and a perimeter term. Our scaling allows low-energy states to be described by an order parameter which can have finitely many point singularities (vortex-like defects) of (possibly) fractional-degree connected by line discontinuities (string defects) of finite length. Our main result is a compactness and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2886_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-convergence theorem which shows how the coarse grained limit energy depends on the geometry of the manifold in driving the interaction between vortices and string defects.</p>

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Renormalized energy between fractional vortices with topologically induced free discontinuities on 2-dimensional Riemannian manifolds

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  • 发表日期：

    2025年2月17日

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  • Rufat Badal,
  • Marco Cicalese

摘要

On a two-dimensional Riemannian manifold without boundary we consider the variational limit of a family of functionals given by the sum of two terms: a Ginzburg–Landau and a perimeter term. Our scaling allows low-energy states to be described by an order parameter which can have finitely many point singularities (vortex-like defects) of (possibly) fractional-degree connected by line discontinuities (string defects) of finite length. Our main result is a compactness and \(\Gamma \) Γ -convergence theorem which shows how the coarse grained limit energy depends on the geometry of the manifold in driving the interaction between vortices and string defects.