Abstract. We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B 1 ⊂ R 2 lies in the local Hardy space h 1 (B 1 ). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B 1 .
We define a new modified mean curvature flow (MMCF) in hyperbolic space H n+1 , which interestingly turns out to be the natural negative L 2 -gradient flow of the energy functional introduced by De Silva and Spruck in [DS09]. We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with prescribed asymptotic boundary at infinity. The proof of our main theorems follows closely Guan and Spruck's work [GS00], and may be thought of as a parabolic analogue.
In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in R n to the Riemannian setting.More precisely, we generalize the min-max construction of minimal surfaces using harmonic replacement introduced by Colding and Minicozzi to the free boundary setting. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit 2-disk in R 2 into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit 2-disk proved by Colding and Minicozzi.
We show an energy convexity along any harmonic map heat flow with small
initial energy and fixed boundary data on the unit 2-disk. In particular, this
gives an affirmative answer to a question raised by W. Minicozzi asking whether
such harmonic map heat flow converges uniformly in time strongly in the
W^{1,2}-topology, as time goes to infinity, to the unique limiting harmonic
map.Comment: 19 page
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