By refined volume estimates in terms of Ricci curvature, the two results due to J. Milnor (1968) are generalized.

Ricci solitons are natural generalizations of Einstein metrics on one hand, and are special solutions of the Ricci flow of Hamilton on the other hand. In this paper we survey some of the recent developments on Ricci solitons and the role they play in the singularity study of the Ricci flow.

In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold (M, g) with Ric (M) ≥ -(n - 1) and Diam (M) ≤ D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups ${\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{H}^{*} {\left( M \right)},{\left\| \cdot \right\|}_{\infty } } \right)}$ among closed Riemannian manifold (M, g) with K(M) ≥ - 1 and Diam (M) ≤ D.

We identify ℝ⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S ⁶. It is known that a cone over a surface M in S ⁶ is an associative submanifold of ℝ⁷ if and only if M is almost complex in S ⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S ⁶ are the equation for primitive maps associated to the 6-symmetric space G ₂=T ², and use this to explain some of the known results. Moreover, the equation for S ¹-symmetric almost complex curves in S ⁶ is the periodic Toda lattice, and a discussion of periodic solutions is given.

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.

This note concerns the global existence and convergence of the solution for Kähler-Ricci flow equation when the canonical class, K _X, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.

A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in ℝ ⁿ⁺¹, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X _n+1 =constant in case M satisfies: for any two points (X′,X _n+1), ${\left( {{X}\ifmmode{'}\else$'$\fi,\ifmmode\expandafter\hat\else\expandafter\^\fi{X}_{{n + 1}} } \right)}$ on M, with $X_{{n + 1}} > \ifmmode\expandafter\hat\else\expandafter\^\fi{X}_{{n + 1}} $, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems.

We rephrase the Gopakumar-Vafa conjecture on genus zero Gromov-Witten invariants of Calabi-Yau threefolds in terms of the virtual degree of the moduli of pure dimension one stable sheaves and investigate the conjecture for K3 fibred local Calabi-Yau threefolds.