Related papers: Collapsed manifolds with local Ricci bounded cover…

Let $M$ be a compact $n$-manifold of $\operatorname{Ric}_M\ge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following…

We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds $\{(M_\alpha^n,g_\alpha)\}_{\alpha \in A}$ whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet…

We study orientability in spaces with Ricci curvature bounded below. Building on the theory developed by Honda, we establish equivalent characterizations of orientability for Ricci limit and RCD spaces in terms of the orientability of their…

We study Riemannian metrics on compact, torsionless, non-geometric $3$-manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "\`a la Margulis" for the systole and a volume estimate for…

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old…

We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple…

Let $M$ be a finite volume analytic pseudo-Riemannian manifold that admits an isometric $G$-action with a dense orbit, where $G$ is a connected non-compact simple Lie group. For low-dimensional $M$, i.e. $\dim(M) < 2\dim(G)$, when the…

We study the geometric properties of a $(2m+1)$-dimensional complex manifold $\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset \mathrm{Spin}(2m+1,\mathbb{C})$, the stabiliser of the line…

This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the…

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it…

We formulate stable Bernstein type theorems in certain positively curved ambient manifolds. In all dimensions, we prove that for any complete Riemannian manifold $(X^{n+1},g)$, if the Ricci curvature is non-negative and it positive BiRic…

In this paper, we construct local and global solutions to the K\"ahler-Ricci flow from a non-collapsed K\"ahler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the…

We investigate the compact submanifolds in Riemannian space forms of nonnegative sectional curvature that satisfy a lower bound on the Ricci curvature, that bound depending solely on the length of the mean curvature vector of the immersion.…

In this paper we introduce a new approach to variational problems on the space Riem(M^n) of Riemannian structures (i.e. isometry classes of Riemannan metrics) on any fixed compact manifold M^n of dimension n >= 5. This approach often…

We study the existence of $L^2$ holomorphic sections of invariant line bundles over Galois coverings of Zariski open sets in Moishezon manilolds. We show that the von Neuman dimension of the space of $L^2$ holomorphic sections is bounded…

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.

In this paper, we give the full proof of a conjecture of R.Hamilton that for $(M^3, g)$ being a complete Riemannian 3-manifold with bounded curvature and with the Ricci pinching condition $Rc\geq \ep R g$, where $R>0$ is the positive scalar…

We survey all results concerning the topology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature that have no additional conditions other than restrictions to the dimension, volume growth or diameter growth of the…

We prove a version of Gromov's compactness theorem for quasiregular curves into calibrated manifolds with bounded geometry. In our main theorem, given an $n$-dimensional calibration $\omega$ on manifold $N$, we associate to a weak-$\star$…

On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real…