Related papers: Combinatorial $p$-th Calabi flows on surfaces

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary \cite{GL2}, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere…

We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in $\mathbb{R}^3$. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining…

Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$. Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $\int_{\partial P} u ~ d \sigma < C_2, $ then there exists a…

Combinatorial Ricci flow on an ideally triangulated compact 3-manifold with boundary was introduced by Luo as a 3-dimensional analog of Chow-Luo's combinatorial Ricci flow on a triangulated surface and conjectured to find algorithmically…

Let X be a toric surface with Delzant polygon P and u(t) be a solution of the Calabi flow equation on P. Suppose the Calabi flow exists in [0, T). By studying local estimates of the Riemann curvature and the geodesic distance under the…

We consider a general curvature equation $F(\kappa)=G(X,\nu(X))$, where $\kappa$ is the principal curvature of the hypersurface $M$ with position vector $X$. It includes the classical prescribed curvature measures problem and area measures…

We show that the analog of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle…

This paper investigates a kind of degenerated circle packings in hyperbolic background geometry. A main problem is whether a prescribed total geodesic curvature data can be realized by a degenerated circle packing or not. We fully…

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric decomposition into ideal hyperbolic tetrahedra, a result proven only for certain special 3-manifolds. This paper presents combinatorial Ricci…

Suppose there is a constant scalar curvature metric on a compact Kahler manifold without holomorphic vector field. We prove that the Calabi flow, if it is assumed to exist for all time with bounded Ricci curvature, will converge to the…

We study the convergence and curvature blow up of La Nave and Tian's continuity method on a generalised Hirzebruch surface. We show that the Gromov-Hausdorff convergence is similar to that of the Kahler-Ricci flow and obtain curvature…

This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The…

We consider the formation of singularities along the Calabi flow with the assumption of the uniform Sobolev constant. In particular, on K\"ahler surface we show that any "maximal bubble" has to be a scalar flat ALE K\"ahler metric. In some…

We make several improvements on the results of M.-T. Wang in [8] and his joint paper with M.-P. Tsui [7] concerning the long time existence and convergence for solutions of mean curvature flow in higher co-dimension. Both the curvature…

Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $ \int_{\partial P} u ~ d \sigma < C_1, $ then there exists a constant…

In this short note we prove that if the curvature tensor is uniformly bounded along the Calabi flow and the Mabuchi energy is proper, then the flow converges to a constant scalar curvature metric.

In this paper, we introduce combinatorial Ricci flows (CRFs in short) in Euclidean and hyperbolic background geometries on infinite triangulations of the open disk, which are discrete analogs of Ricci flows on simply connected open…

We consider the inverse curvature flows $\dot x=F^{-p}\nu$ of closed star-shaped hypersurfaces in Euclidean space in case $0<p\not=1$ and prove that the flow exists for all time and converges to infinity, if $0<p<1$, while in case $p>1$,…

In this paper, we prove the long time regularity of the interface in the $p$-Gauss curvature flow with flat side in all dimensions for $p>\frac1n$. Here the interface is the boundary of the flat part in the flow. In dimension $2$, this…

We generalize Tutte's integer flows and the $d$-dimensional Euclidean flows of Mattiolo, Mazzuoccolo, Rajn\'{i}k, and Tabarelli to \emph{$d$-dimensional $p$-normed nowhere-zero flows} and define the corresponding flow index $\phi_{d,p}(G)$…