We consider an embedded convex compact ancient solution t to the curve shortening flow in R2. We prove that t is either a family of contracting circles, which is a type I ancient solution, or a family of evolving Angenent ovals, which is of type II.

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We show that any star-shaped convex hypersurface with constantWeingarten curvature in the deSitter-Schwarzschildmanifold is a sphere of symmetry.Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of the enclosed volume, and we completely describe the isoperimetric surfaces for very large enclosed volume. This complements work in H. Bray’s thesis, where isoperimetric surfaces homologous to the horizon are studied.

For a one parameter family of Calabi-Yau threefolds, Green et al. (2009) expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV (Bershadsky-Cecotti-Ooguri-Vafa) invariants, studied by Fang et al. (2008). On the other hand, by using Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound for Calabi-Yau family over a compact Riemann surface.

We give a complete diffeomorphism classification of 1-connected closed manifolds M with integral homology H¤(M) ∼= Z ⊕ Z ⊕ Z, provided that dim(M) 6= 4.