We show that Norbury’s McShane identity for nonorientable cusped hyperbolic surfaces N generalizes to quasifuchsian representations of π1(N) as well as pseudo-Anosov mapping Klein bottles with singular fibers given by N.

Let  $\sum $ be a compact oriented surface immersed in a four dimensional K¨ahler-Einsteinmanifold $(M, ω)$. We consider the evolution of $\sum$ in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When $M$ has two parallel K¨ahler forms $\omega " $ and $\omega ''$ that determine different orientations and  $\sum$ is symplectic with respect to both $\omega " $ and $\omega ''$, we prove the mean curvature flow of $\sum$  exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.

Let g(t) be a smooth complete solution to the Ricci flow on a noncompact manifold such that g(0) is Kahler. We prove that if |Rm(g(t))| is bounded by a/t for some a > 0, then g(t) is Kahler for t > 0. We prove that there is a constant a(n) > 0 depending only on n such that the following is true: Suppose g(t) is a smooth complete solution to the Kahler-Ricci flow on a non-compact n-dimensional complex manifold such that g(0) has nonnegative holomorphic bisectional curvature and |Rm(g(t))| ≤ a(n)/t, then g(t) has nonnegative holomorphic bisectional curvature for t > 0. These generalize the results by Wan-Xiong Shi. As applications, we prove that (i) any complete noncompact Kahler manifold with nonnegative complex sectional curvature and maximum volume growth is biholomorphic to C^n; and (ii) there is ε(n) > 0 depending only on n such that if (M^n,g_0) is a complete noncompact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature and maximum volume growth and if (1+ε(n))^{−1}h ≤ g_0 ≤ (1+ε(n))h for some Riemannian metric h with bounded curvature, then M is biholomorphic to C^n.

In this paper, we prove a scalar curvature rigidity result for geodesic balls in Sn. This result contrasts sharply with the counterexamples to Min-Oo’s conjecture constructed in [7].

We give a natural filtration F on QH(G/B), which respects the quantum product structure. Its associated graded algebra GrF(QH(G/B)) is isomorphic to the tensor product of QH(G/P) and a corresponding graded algebra of QH(P/B) after localization. When the quantum parameter goes to zero, this specializes to the filtration on H(G/B) from the Leray spectral sequence associated to the fibration P/B ! G/B −! G/P.