arXiv subjects
Publications and source records attributed to Xilun Li.
We prove the Kobayashi-Hitchin correspondence for parabolic bundles over compact nonK\"{a}hler surfaces with simple normal crossing divisor or compact nonK\"{a}hler manifolds of any dimension with smooth divisor.
For any integers $m\geqslant n\geqslant 3$, we construct a Ricci limit space $X_{m,n}$ such that for a fixed point, some tangent cones are $\mathbb{R}^m$ and some are $\mathbb{R}^n$. This is an improvement of Menguy's example. Moreover, we show that for any finite collection of closed Riemannian manifolds $(M_i^{n_i},g_i)$ with $\mathrm{Ric}_{g_i}\geqslant(n_i-1)\geqslant 1$, there exists a collapsed Ricci limit space $(X,d,x)$ such that each Riemannian cone $C(M_i,g_i)$ is a tangent cone of $X$ at $x$.
We show that every gradient shrinking soliton of the generalized Ricci flow on compact manifold is a Ricci soliton. And we prove that the pluriclosed soliton is gradient Kahler-Ricci soliton under a broad cohomological condition. Moreover, we construct the first example of non-trivial shrinking generalized soliton, which can serve as a singularity model of the generalized Ricci flow.
We introduce analogous geometric quantities and prove some geometric and analytic bounds in [Bam20a] to generalized Ricci flow.