An improved uniqueness result for the harmonic map flow in two dimensions

Generalizing a result of Freire regarding the uniqueness of the harmonic map flow from surfaces to an arbitrary closed target manifold N, we show uniqueness of weak solutions u ∈ H ¹ under the assumption that any upwards jumps of the energy function are smaller than a geometrical constant , thus establishing a conjecture of Topping, under the sole additional condition that the variation of the energy is locally finite.