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Abstract
Abstract The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as t → ∞ {t\to\infty} .
Highlights
Introduction and resultsIn their seminal work [2] from 1964, Eells and Sampson showed that the harmonic map flow into a closed manifold of nonpositive sectional curvature admits global solutions starting with arbitrary smooth initial data, which converge to a limiting harmonic map when restricted to a suitable sequence of times ti → ∞
In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms
We investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as t → ∞
Summary
In their seminal work [2] from 1964, Eells and Sampson showed that the harmonic map flow into a closed manifold of nonpositive sectional curvature admits global solutions starting with arbitrary smooth initial data, which converge to a limiting harmonic map when restricted to a suitable sequence of times ti → ∞. There exist a closed target manifold and a solution of Teichmüller harmonic map flow from T2 into that target for which the metric component stays in a compact subset of Teichmüller space, but for which the limit as t → ∞ is not unique. Since the initial map is incompressible, this suffices to apply the argument of [1, Remark 4.4] in the case of a non-compact target manifold to conclude that the injectivity radius of (T2, g) remains bounded from below by a positive constant c > 0 that only depends on the initial energy and the curve Gs. We note that (2.6) holds true on the maximal time interval [0, T) as an immediate consequence of Lemma 2.3, the structure of the energy (2.2) and the fact that f0 ≥ 1.
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