This result states convexity of energy along geodesic homotopies into nonpositively curved targets: given (M,g) be a compact smooth Riemannian manifold without boundary and let (N,h) have sectional curvature K N 0. Let U:M [0,1] N be a smooth geodesic homotopy, meaning that for.... It is useful in harmonic map theory and elliptic regularity, where variational identities, curvature estimates, and compactness arguments control geometric objects.