Curvature flows toward optimal geometric inequalities

Solutions to many problems in geometry are often characterized by "optimal" inequalities. The classical isoperimetric inequality is such an example. A curve shorting flow proof by Gage and Hamilton for the classical perimetric inequality illustrated the role of curvature flow. Geometric quantities are often associated with important variational properties, and these properties enable us to design curvature flows as "good paths" descending to the optimal geometric inequalities. For example, a new mean curvature type flow naturally emerging in the case of isoperimetric problem. This idea can be adapted to establish new geometric inequalities. (The talk is for general audience.)