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<p>We show that Legendrian integral currents in a contact manifold that locally minimize the mass among Legendrian competitors have a regular set which is open and dense in their support. We apply this to show the existence and partial regularity of solutions of the Legendrian Plateau problem in the <i>n</i>th Heisenberg group for an arbitrary horizontal <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((n-1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-cycle as prescribed boundary, and of mass-minimizing Legendrian integral currents in any <i>n</i>-dimensional homology class of a closed contact <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((2n+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-manifold. In the case of the Heisenberg group, our result applies to Ambrosio–Kirchheim metric currents with respect to the Carnot–Carathéodory distance. Our results do not assume any compatibility between the subriemannian metric and the symplectic form.</p>

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Existence and Partial Regularity of Legendrian Area-Minimizing Currents

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  • 发表日期：

    2026年7月2日

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  • Gerard Orriols

摘要

We show that Legendrian integral currents in a contact manifold that locally minimize the mass among Legendrian competitors have a regular set which is open and dense in their support. We apply this to show the existence and partial regularity of solutions of the Legendrian Plateau problem in the nth Heisenberg group for an arbitrary horizontal \((n-1)\) ( n - 1 ) -cycle as prescribed boundary, and of mass-minimizing Legendrian integral currents in any n-dimensional homology class of a closed contact \((2n+1)\) ( 2 n + 1 ) -manifold. In the case of the Heisenberg group, our result applies to Ambrosio–Kirchheim metric currents with respect to the Carnot–Carathéodory distance. Our results do not assume any compatibility between the subriemannian metric and the symplectic form.