Singularities of first kind in the harmonic map and Yang-Mills heat flows

Abstract

By constructing examples which are explicit up to solving an ODE, we prove that singularities of first kind exist for the harmonic map and Yang-Mills heat flows. As a by-product, we also get a simplified proof of Ratto's/Ding's theorem about the existence of harmonic Hopf constructions. Singularities of first kind in the harmonic map heat flowLet M m , N n ⊂ IR n+k be compact Riemannian manifolds. We say that u : [0, T ) × M → N is a solution of the harmonic map heat flow if it satisfies the system of differential equationswhere ∆ M is the Laplace-Beltrami operator of M and A N the (vectorvalued) second fundamental form of N ; Du denotes the total derivative of u with respect to the space variables. This flow was used in the fundamental work [ES] of Eells and Sampson to prove existence of harmonic maps to manifolds of nonpositive sectional curvature. They actually proved that, given smooth initial data u(0, · ) = u 0 : M → N , the harmonic map heat flow has a smooth solution and f := lim i→∞ u(t i , · ) defines a harmonic map (homotopic to u 0 via the flow) for a suitable sequence t i → ∞. This may fail if N has positive sectional curvatures somewhere. Actually, there exists T > 0 such that u is smooth on [0, T ) × M , and if T is maximal, then lim t→T sup M |Du(t, x)| = ∞, in which case we say that u blows up at t = T .Thus a natural question is: Does blow-up happen at finite time, and can we understand the nature of the singularity? That blow-up at finite time

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