The Willmore Conjecture

The Willmore Conjecture concerns the shape of surfaces like soap bubbles or biological membranes. It states that among all tori-shaped surfaces (donuts), the one with the smallest total bending energy—called the Willmore energy—is a specific, perfectly symmetric shape. This "optimal" shape minimizes the surface's bending without stretching or tearing. The conjecture, proven in 2012, shows that the simplest donut shape (the Clifford torus) uniquely minimizes this energy, providing insight into the geometry of surfaces and how they naturally minimize bending.