Abstract:
"The ball is the only domain in Euclidean space that minimizes area for a given volume." This simple statement encapsulates the full essence and power of geometric inequalities: the (scale-invariant) ratio between an object's area and volume is always bounded from below by a dimensional constant. By measuring it, I can determine whether the object is a ball.
But what if the space around us is curved? And what if we consider other geometric quantities, such as the mean curvature? Through some selected examples, I will introduce a powerful PDE-based technique that has been successfully applied to address these questions.