Date: 19 Mar 2008

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

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Abstract

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 \(\epsilon^\star=\epsilon^\star(N)\) , thus establishing a conjecture of Topping, under the sole additional condition that the variation of the energy is locally finite.