We consider the inverse curvature flow of strictly convex hypersurfaces in the space form N of constant sectional curvature K_N with speed given by F^{−α}, where α∈(0,1] for K_N=0,−1 and α=1 for K_N=1, F is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual F_∗ approaching zero on the boundary of the positive cone Γ_+. We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flows.

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We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on\cite {Th}. We give new results about the stability condition, and propose a Jordan-Hlder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's examples of SLags.

We consider a 1-parameter family of strictly convex hypersurfaces in Rn+1 moving with speed −Kαν, where ν denotes the outward-pointing unit normal vector and α⩾1/(n+2). For α>1/(n+2), we show that the flow converges to a round sphere after rescaling. In the affine invariant case α=1/(n+2), our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.