Willmore surfaces

Willmore surfaces are special types of smooth shapes in geometry that minimize a specific bending energy called the Willmore energy. Think of them as the most "efficiently shaped" surfaces when it comes to bending, similar to how soap bubbles naturally form minimal surfaces to reduce surface tension. These surfaces are critical points of the Willmore energy functional, meaning they are locally optimal shapes that balance bending without necessarily being perfect spheres. They are studied in differential geometry for their elegant properties and appear in theories related to physics, biology, and material science, helping us understand natural and engineered shapes.