This result states parabolic bochner inequality for harmonic map heat flow into nonpositively curved targets: given M be closed and let u:M [0,T) N be a smooth solution of harmonic map heat flow. If N has nonpositive sectional curvature, then there is a constant C M 0, depending only on.... It is useful in harmonic map theory and elliptic regularity, where variational identities, curvature estimates, and compactness arguments control geometric objects.