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A uniform Poincaré estimate for quadratic differentials on closed surfaces
- In preparation
"... Abstract. We revisit the classical Poincaré inequality on closed surfaces, and prove its natural analogue for quadratic differentials. In stark contrast to the classical case, our inequality does not degenerate when we work on hyperbolic surfaces that themselves are degenerating, and this fact turns ..."
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Cited by 5 (5 self)
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Abstract. We revisit the classical Poincaré inequality on closed surfaces, and prove its natural analogue for quadratic differentials. In stark contrast to the classical case, our inequality does not degenerate when we work on hyperbolic surfaces that themselves are degenerating, and this fact turns out to be essential for applications to the Teichmüller harmonic map flow. 1.
Flowing maps to minimal surfaces
"... We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic ma ..."
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Cited by 3 (3 self)
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We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow, and it tries to find branched minimal 2-spheres as in Sacks-Uhlenbeck [10] and Struwe [13] etc. In the genus 1 case, we show that our flow is exactly equivalent to that considered by Ding-Li-Lui [1]. In general, we recover the result of Schoen-Yau [12] and Sacks-Uhlenbeck [11] that an incompressible map from a surface can be adjusted to a branched minimal immersion with the same action on π1, and this minimal immersion will be homotopic to the original map in the case that π2 = 0.
Refined asymptotics of the Teichmüller harmonic map flow into general targets
"... Abstract The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for ..."
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Abstract The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for all time t ≥ 0, we find a sequence of times ti → ∞ at which the flow at different scales converges to a collection of branched minimal immersions with no loss of energy. We do this by developing a compactness theory, establishing no loss of energy, for sequences of almost-minimal maps. Moreover, we construct an example of a smooth flow for which the image of the limit branched minimal immersions is disconnected. In general, we show that the necks connecting the images of the branched minimal immersions become arbitrarily thin as i → ∞.
