Related papers: Ricci flow on surfaces with conical singularities

This is the fourth and last part of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly,…

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset H(q,n) of the variety of (q+n)-dimensional Lie algebras, parameterizing the space of all…

We survey several problems concerning Riemannian manifolds with positive curvature of one form or another. We describe the PIC1 notion of positive curvature and argue that it is often the sharp notion of positive curvature to consider.…

In this article, we establish a monotonicity formula of Hamilton type entropy along Ricci flow on compact surfaces with boundary. We also study the relation between our entropy functional and the $\mathcal{W}$-functional of Perelman type.

The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly.…

We describe the Ricci flow on two classes of compact three-dimensional manifolds: 1. Warped products with a circle fiber over a two-dimensional base. 2. Manifolds with a free local isometric U(1) x U(1) action.

This paper has been withdrawn by the author for further modification.

We extend some results known for the K\"ahler-Ricci flow to the Chern-Ricci flow regarding the independence of singularity types for long-time solutions. Specifically, we show that if a solution to the Chern-Ricci flow exists with uniformly…

We present local estimates for solutions to the Ricci flow, without the assumption that the solution has bounded curvature. These estimates lead to a generalisation of one of the pseudolocality results of G.Perelman in dimension two.

We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic…

We define the Ricci curvature, as a measure, for certain singular torsion-free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the…

We establish the existence of the K"ahler-Ricci flow on projective varieties with log canonical singularities. This generalizes some of the existence results of Song-Tian \cite{ST3} in case of projective varieties with klt singularities. We…

We study the uniqueness problem for the K\"ahler-Ricci flow with a conical initial condition. Given a complete gradient expanding K\"ahler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we…

We study monotonic quantities in the context of combined geometric flows. In particular, focusing on Ricci solitons as the ambient space, we consider solutions of the heat type equation integrated over embedded submanifolds evolving by mean…

We present two characterizations of smooth compact Ricci flow solutions solely in terms of metrics and measures (one of them only works under positive scalar curvature along the flow); thus, provide weak formulations that are generalized to…

We can talk about two kinds of stability of the Ricci flow at Ricci flat metrics. One of them is a linear stability, defined with respect to Perelman's functional $\mathcal{F}$. The other one is a dynamical stability and it refers to a…

In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In…

In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton's Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and…

In this work, we study the short-time existence theory of Ricci-DeTurck flow starting from rough metrics which satisfy a Morrey-type integrability condition. Using the rough existence theory, we show the preservation and improvement of…

We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on surfaces. We show that existence for the flow reduces to a bound on the isoperimetric constant. In the presence of such a bound, we show that on $S^2$, if the bundle is…