Related papers: Ancient Solutions on Bundles with Non-abelian Stru…

We consider compact ancient solutions to the three-dimensional Ricci flow which are noncollapsed. We prove that such a solutions is either a family of shrinking round spheres, or it has a unique asymptotic behavior as $t \to -\infty$ which…

In [4], we proved that every noncompact ancient $\kappa$-solution to the Ricci flow in dimension $3$ is either locally isometric to a family of shrinking cylinders, or isometric to the Bryant soliton. In the same paper, we announced that…

It is well known that the K\"ahler-Ricci flow on a K\"ahler manifold $X$ admits a long-time solution if and only if $X$ is a minimal model, i.e., the canonical line bundle $K_X$ is nef. The abundance conjecture in algebraic geometry…

We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as…

We produce new examples of non-K\"ahler complete expanding gradient Ricci solitons on trivial vector bundles over a product of Einstein manifolds with positive scalar curvature.

In this paper, we consider the classification of compact ancient noncollapsed mean curvature flows of hypersurfaces in arbitrary dimensions. More precisely, we study $k$-ovals in $\mathbb{R}^{n+1}$, defined as ancient noncollapsed solutions…

In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature bound. New results on ancient…

We show that any manifold admitting a non-collapsed, ancient Ricci flow must have finite fundamental group. This generalizes what was known for $\kappa$-solutions in dimensions 2, 3. We furthermore show that this fundamental group must be a…

We survey some recent developments on solutions of the K\"ahler-Ricci flow on compact K\"ahler manifolds which exist for all positive times.

In this work, we study the H\"older regularity of the K\"ahler- Ricci flow on compact K\"ahler manifolds with semi-ample canonical line bundle. By adapting the method in the work of Hein-Tosatti on collapsing Calabi-Yau metrics, we…

Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized K\"ahler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure…

We find explicit solutions and singularities of the Ricci-harmonic flow of $\mathrm{G_2}$-structures, the Ricci-like flows of $\mathrm{G_2}$-structures studied by Gianniotis-Zacharopoulos in arXiv:2505.06872 (J. Geom. Anal. 36.2 (2026)) and…

Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact K\"ahler manifolds. In this short note, we give anothe proof of…

In this article, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the…

The main result of this paper is: Given any constant C, there is $(\epsilon,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(\epsilon,k,L)$-neck, then the…

We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the…

We produce complete bounded curvature solutions to K\"ahler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain…

We show that any non-collapsed finite time singularity of the Ricci flow on a compact K\"ahler surface is of Type I. Combined with a previous result of the first author, Cifarelli, and Deruelle, it follows that any such singularity is…

We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the…

We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact K\"ahler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of…