Related papers: A normalized Ricci flow on surfaces with boundary …

We establish the existence of K\"ahler-Ricci flow on pseudoconvex domains with general initial metric without curvature bounds. Moreover we prove that this flow is simultaneously complete, and its normalized version converge to the complete…

Let M be a compact n-dimensional manifold, $n\ge 2$, with metric g(t) evolving by the Ricci flow $\partial g_{ij}/\partial t=-2R_{ij}$ in (0,T) for some $T\in\Bbb{R}^+\cup\{\infty\}$ with $g(0)=g_0$. Let $\lambda_0(g_0)$ be the first…

In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has…

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as…

We describe a nonsmooth notion of globally hyperbolic, regular length metric spacetimes $(\mathrm{M},l)$. It is based on ideas of Kunzinger-S\"amann, but does not require Lipschitz continuity of causal curves. We study geodesics on…

We establish an estimate for the fundamental solution of the heat equation on a closed Riemannian manifold $M$ of dimension at least 3, evolving under the Ricci flow. The estimate depends on some constants arising from a Sobolev imbedding…

This paper is concerned with properties of maximal solutions of the Ricci and cross curvature flows on locally homogeneous three-manifolds of type SL(2,R). We prove that, generically, a maximal solution originates at a sub-Riemannian…

In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower…

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…

In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the initial data a unique…

We consider smooth, not necessarily complete, Ricci flows, $(M,g(t))_{t\in (0,T)}$ with ${\mathrm{Ric}}(g(t)) \geq -1$ and $| {\mathrm{Rm}} (g(t))| \leq c/t$ for all $t\in (0 ,T)$ coming out of metric spaces $(M,d_0)$ in the sense that…

The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the…

In this paper we establish new geometric and analytic bounds for Ricci flows, which will form the basis of a compactness, partial regularity and structure theory for Ricci flows in [Bam20a, Bam20b]. The bounds are optimal up to a constant…

Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by Gurtin and Podio-Guidugli (1991) to model certain wave phenomena in…

On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a metric of constant curvature and, for…

The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface,…

We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as special planar dynamical systems. All non symmetric generalized Wallach spaces can be naturally parametrized by three…

We prove that, given $\alpha>0$, if $M$ is a complete Riemannian manifold which Ricci curvature satisfies.\[\operatorname*{Ric}\nolimits_{x}(v)\geq\alpha\operatorname{sech}^{2} (r(x)))\] or \[…

In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed $n$-dimensional K\"ahler Ricci flow. These integral estimates improve…

In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat…