Related papers: Margulis Lemma on $\text{RCD}(K,N)$ spaces

In this paper we present several curvature estimates for solutions of the Ricci flow which depend on smallness of certain local integrals of the norm of the Riemann curvature tensor.

We present two new conditions to extend the Ricci flow on a compact manifold over a finite time, which are improvements of some known extension theorems.

In this paper we investigate the Margulis-Ruelle inequality for general Riemannian manifolds (possibly noncompact and with boundary) and show that it always holds under integrable condition.

The main objective of this article is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic…

We prove that Ahlfors-regular RCD spaces are uniformly rectifiable. The same is shown for Ahlfors regular boundaries of non-collapsed RCD spaces. As an application we deduce a type of quantitative differentiation for Lipschitz functions on…

We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition,…

We prove that a Ricci flow cannot develop a finite time singularity assuming the boundedness of a suitable space-time integral norm of the curvature tensor. Moreover, the extensibility of the flow is proved under a Ricci lower bound and the…

In this short note we announce a regularity theorem for K\"ahler-Ricci flow on a compact Fano manifold (K\"ahler manifold with positive first Chern class) and its application to the limiting behavior of K\"ahler-Ricci flow on Fano…

For a general $k$-dimensional Brakke flow in $\mathbb{R}^n$ locally close to a $k$-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the…

In this article, we prove an $\epsilon$-regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to $1$, then the curvature radius at the base point cannot be too small.

Following work of Ecker, we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman's modified Ricci flow. The answer has a…

We show that a complete Ricci flow of bounded curvature which begins from a manifold with a Ricci lower bound, local entropy bound, and small local scale-invariant integral curvature control will have global point-wise curvature control at…

Assume $M$ is a closed 3-manifold whose universal covering is not $S^3$. We show that the obstruction to extend the Ricci flow is the boundedness $L^{3/2}$-norm of the scalar curvature $R(t)$, i.e, the Ricci flow can be extended over time…

We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local…

In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean…

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $ \phi_\lambda $ on a closed Riemannian manifold $ (M,g) $. First, in the real-analytic case we present an improvement of…

In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the…

In this paper, we extend the sharp lower bounds of spectal gap, due to Chen- Wang [10, 11], Bakry-Qian [6] and Andrews-Clutterbuck [5], from smooth Riemaniannian manifolds to general metric measure spaces with Riemannian curvature-dimension…

In this note, we provide some general discussion on the Ricci lower bound along K\"ahler-Ricci flow with singularity over closed manifold.

On a fairly general class of Riemannian manifolds M, we prove lower estimates in terms of the Ricci curvature for the spectral bound (when M has infinite volume) and for the spectral gap (when M has finite volume) for the Laplace-Beltrami…