Somebody tolds me that the solution of variational problems in smooth manifold can provide us some topological information of the underlying manifold.
For example, let $(M,g)$ be a compact riemannian manifold and $\Delta^r:\Omega^r(M)\to\Omega^r(M)$ the Laplace-Beltrami operator of $r$-forms (a natural generalization of the Laplace operator in $\mathbb R^n$). It may be surprising that the deRham cohomology classes $H^r(M)$ are isomorphic to the kernel $\ker \Delta^r$ (by the famous Hodge Theorem).
This implies that the number of harmonic forms (elements of $\omega$) provide us global topological information of $M$!
It can be shown that the harmonic forms are the critical points of the Lagrangian $L:\Omega^r(M)\to \mathbb R$, $$L(\alpha)=\int_M\tilde g(\alpha,\alpha)\, \xi_g,$$ where $\xi_g$ is a volume form in $M$ ant $\tilde g$ is a cometric induced in $\Omega^r(M)$. In this view, the harmonic form are the ''extremals'' for the action functional $L$.
I was wondering if there is any reference where I can find more applications of variational problems to topology.
Thanks in advances.