Eells and Sampson established the existence of harmonic maps whenever the domain manifold is closed and the target has non-positive sectional curvature.
The use of the Bochner identity in deriving estimates is where the assumption on sectional curvature plays a crucial role.
As a result of Eells and Sampson's (subsequential) convergence theorem, they were able to prove the existence of harmonic maps in any homotopy class.
In addition to Eells and Sampson's heat flow, their main results on existence of harmonic maps can also be derived via the calculus of variations, using the regularity theory developed in the 1980s by Richard Schoen and Karen Uhlenbeck.
Later in 1978, Sampson developed unique continuation, maximum principles, further rigidity theorems, and deformability results for harmonic maps.