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Existence of good sweepouts on closed manifolds
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https://doi.org/10.1090/s0002-9939-2010-10451-3Abstract
In this note we establish estimates for the harmonic map heat flow from $S^1$ into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic.
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