Energies of geometric objects under infinitesimal deformation (or bending) measure how curvature-based shape functionals—like Willmore or torsional energy—change. Because infinitesimal bending preserves intrinsic geometry (metric and Gaussian curvature).

The relationship between the shape of geometric objects and their energies is a fundamental concept that explains their properties and behavior in various fields, including nature, engineering, and architecture. Energy-minimizing shapes, like spheres for droplets or spherical cells are common because they are efficient and stable. Applications range from designing energy-absorbing thin-walled structures to understanding the properties, demonstrating the universal principle that form follows function and energy optimization.

In this talk we will consider specially shape and energy under infinitesimal bending.

Reference

Shape and Energies of Geometric Objects

Louis H Kauffman (University of Illinois at Chicago, USA) Ljubica S Velimirović (University of Niš, Serbia), Marija S Najdanović (University of Priština in Kosovska Mitrovica, Serbia), and Svetozar R Rančić (University of Niš, Serbia)

Series on Knots and Everything: Volume 80

World Scientific, April 2026