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<p>An exact differential two-form is constructed in the injective hull of the Riemannian circle, whose comass norm–defined via the inscribed Riemannian area on normed planes–is stationary at every point of the open hemisphere spanned by the circle. As a consequence, in any metric space, the induced Finsler mass of a two-dimensional Ambrosio-Kirchheim rectifiable current with boundary a Riemannian circle of length <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> admits a lower bound of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(2\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> plus a second-order term in the Hausdorff distance to an isometric copy of the hemisphere. This estimate applies to all oriented Lipschitz surfaces spanning the circle, regardless of their topology, and thus offers positive evidence for Gromov’s filling area conjecture.</p>

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The Riemannian hemisphere is almost calibrated in the injective hull of its boundary

  • 期刊论文

  • 发表日期：

    2025年10月15日

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  • Roger Züst

摘要

An exact differential two-form is constructed in the injective hull of the Riemannian circle, whose comass norm–defined via the inscribed Riemannian area on normed planes–is stationary at every point of the open hemisphere spanned by the circle. As a consequence, in any metric space, the induced Finsler mass of a two-dimensional Ambrosio-Kirchheim rectifiable current with boundary a Riemannian circle of length \(2\pi \) 2 π admits a lower bound of \(2\pi \) 2 π plus a second-order term in the Hausdorff distance to an isometric copy of the hemisphere. This estimate applies to all oriented Lipschitz surfaces spanning the circle, regardless of their topology, and thus offers positive evidence for Gromov’s filling area conjecture.