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

Motivated by the goal of detecting minimal surfaces in hyperbolic manifolds, we study geometric flows in complete hyperbolic 3-manifolds. In general, the flows might develop singularities at some finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and R. We show that for a large class of closed initial surfaces, which are graphs over the totally geodesic surface Σ, the mean curvature flow exists for all time and converges to Σ. This is among the first examples of converging mean curvature flows starting from closed hypersurfaces in Riemannian manifolds. We also provide calculations for the general warped product setting which will be useful for further works.

In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball {B_{1}\subset\mathbb{R}^{4}} into the sphere {\mathbb{S}^{n}}. In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into {\mathbb{S}^{n}}, which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm [26]. Further, we establish the previously unknown result that the energy convexity along the flow yields uniform convergence of the flow.

Abstract. It is conjectured that the mean curvature blows up at the first singular time of the mean curvature flow in Euclidean space, at least in dimensions less or equal to 7. We show that the mean curvature blows up at the singularities of the mean curvature flow starting from an immersed closed hypersurface with small L 2 -norm of the traceless second fundamental form (observe that the initial hypersurface is not necessarily convex). As a consequence of the proof of this result we also obtain the dynamic stability of a sphere along the mean curvature flow with respect to the L

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