Longzhi Lin

Estimates for the energy density of critical points of a class of conformally invariant variational problems

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 lies in the local Hardy space . As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B1.

Existence of good sweepouts on closed manifolds

In this note we establish estimates for the harmonic map heat flow from into a closed manifold, and we 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.

Uniformity of harmonic map heat flow at infinite time

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.

Blow-up of the mean curvature at the first singular time of the mean curvature flow

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 than 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

Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

L^2

Mean curvature flow of star-shaped hypersurfaces

L2-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

Mean curvature flow in Fuchsian manifolds

L^2