This is a joint work with Lu Xu. We establish a geometric lower bound for the principal curvature of the level surfaces of solutions to $F(D^2u, Du, u, x)=0$ in convex ring domains, under a refined structural condition introduced by Bianchini-Longinetti-Salani.

We construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches, without performing an intervening surgery. In the restrictive context of rotational symmetry, the construction gives evidence in favor of Perelman's hope for a "canonically defined Ricci flow through singularities". This is joint work with Sigurd Angenent and Cristina Caputo.

I will talk about the rigidity for a local holomorphic isometric embedding
from ${\BB}^n$ into ${\BB}^{N_1} \times\cdots \times{\BB}^{N_m}$ with
respect to the normalized Bergman metrics. Each component of the map is a
multi-valued holomorphic map between complex Euclidean spaces by Mok's
algebraic extension theorem. By using the method of the holomorphic
continuation and analyzing real analytic subvarieties carefully, we show
that a component is either a constant map or a proper holomorphic map
between balls. Hence the total geodesy of non-constant components follows
from a linearity criterion of Huang. In fact, the rigidity is derived in a
more general setting for a local holomorphic conformal embedding. This is
a joint work with Y. Zhang.