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Statics and diffusive dynamics of surfaces driven by p-atic topological defects.

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This study models surface shaping using topological defects, showing positive defects create cones and predicting semi-cone angles. The research reveals deformed lemon shapes in membranes, with implications for epithelial morphogenesis and pollen grain shape transitions.

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Area of Science:

  • Mathematical Biology
  • Soft Matter Physics
  • Surface Geometry

Background:

  • Epithelial morphogenesis involves complex surface shaping processes.
  • Topological defects play a crucial role in biological and physical systems.
  • Understanding the interplay between intrinsic and extrinsic geometry is key.

Purpose of the Study:

  • To develop a minimal model for surface shaping driven by p-atic topological defects.
  • To investigate the dynamic generation and evolution of conical shapes from defects.
  • To predict the final shape and geometric properties of membranes with embedded polar order.

Main Methods:

  • Utilized a minimal mathematical model for surface dynamics.
  • Analyzed the behavior of positive and negative topological defects.
  • Exploited the coupling between extrinsic and intrinsic geometry for axisymmetric surfaces.

Main Results:

  • Positive (negative) defects dynamically generate (hyperbolic) cones with diffusive shape evolution.
  • A defect of charge +1/p predicts a final semi-cone angle β satisfying a specific inequality.
  • Stationary membranes with negligible bending modulus and polar order form deformed lemon shapes with antipodal defects.

Conclusions:

  • The study provides a theoretical framework for defect-driven surface morphogenesis.
  • Results offer insights into the formation of conical and lemon-like surface geometries.
  • Findings may extend to shape transitions in other closed spheroidal surfaces, like pollen grains.