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<p>The Helfrich model is a fundamental tool for determining the morphology of biological membranes. We relate the geometry of an important class of its equilibria to the geometry of sessile and pendant drops in the hyperbolic space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{H}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">H</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. When the membrane surface meets the ideal boundary of the hyperbolic space, a modification of the regularized area functional is related to the construction of closed equilibria for the Helfrich functional in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textbf{R}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">R</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>.</p>

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Hyperbolic Geometry and the Helfrich Functional

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月7日

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  • Bennett Palmer,
  • Álvaro Pámpano

摘要

The Helfrich model is a fundamental tool for determining the morphology of biological membranes. We relate the geometry of an important class of its equilibria to the geometry of sessile and pendant drops in the hyperbolic space \(\textbf{H}^3\) H 3 . When the membrane surface meets the ideal boundary of the hyperbolic space, a modification of the regularized area functional is related to the construction of closed equilibria for the Helfrich functional in \(\textbf{R}^3\) R 3 .