Shijin Zhang

In this short note, using Günther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton $(M^n,g,f)$ with sectional curvature $K(g)<A$ and ${\mathrm{Vol}}_f(M)\ge v$ for some uniform constant $A,v$, there exists a small uniform constant ${\epsilon }_{n,A,v}>0$ depends only on $n,A$ and $v$, if the scalar curvature $R\le {\epsilon }_{n,A,v}$, then $(M,g,f)$ is isometric to the Gaussian soliton $({\mathbb{R}}^n,g_E,\frac{|x{|}^2}{4})$.

In this paper, we prove that any complete shrinking gradient Kähler-Ricci solitons with positive orthogonal bisectional curvature must be compact. We also obtain a classification of the complete shrinking gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature.

In this short note, using an argument by Munteanu and Wang, we provide an alternative proof of the fact first obtained by Lei Ni that shrinking gradient Kähler-Ricci solitons with positive bisectional curvature must be compact.

We study the behavior of the Kähler-Ricci flow on some Fano bundle which is a trivial bundle on one Zariski open set. We show that if the fiber is ${\mathbb{P}}^m$ blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kähler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base.

We prove that there does not exist non-constant positive $f$-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the $L^p(p\ge 1\mathrm{or}0<p\le 1)$ Liouville theorems on the complete gradient shrinking Ricci solitons.