# The heat flow for harmonic maps and applications

In 1964, Eells and Sampson asked whether a given smooth map can be deformed to a harmonic map in its homotopy class. When $n=2$, Lemaire and Schoen-Yau established existence results of harmonic maps by minimizing the Dirichet energy in a homotopic class under the topological condition $\pi_2(N)=0$. Sacks and Uhlenbeck established many existence results of minimizing harmonic maps in their homotopic classes by introducing the `Sacks-Uhlenbeck functional'. To solve the Eells-Sampson question, it is important to establish global existence of a solution of the harmonic map flow. Struwe proved the existence of a unique global weak solution to the harmonic map flow. Chang, Ding and Ye constructed an example that the harmonic map flow blows up at finite time. Ding and Tian established the energy identity of the harmonic map flow at each blow-up time.

When $n>2$, we also introduce some new result on a $n$-harmonic map flow from an n-dimensional closed Riemannian manifold to another closed Riemannian manifold. Finally, we mention some applications of the n-harmonic map flow to minimizing the n-energy functional and the Dirichlet energy functional in a homotopic class.