The Ricci flow, introduced by Richard S. Hamilton, evolves a Riemannian metric in a way that can be thought of as a heat equation for the metric tensor, with the Ricci curvature playing a role analogous to the Laplace-Beltrami operator. This process, which played a key role in Grigori Perelman’s proof of the Poincaré conjecture, further underscores the deep connections between the operator, curvature, and the topology of manifolds.
These examples illustrate how the Laplace-Beltrami operator serves as a bridge between differential geometry and various branches of mathematics and physics, revealing the geometric and topological structure of manifolds through analytical means.
with the Ricci curvature playing a role analogous to the Laplace-Beltrami operator