No Arabic abstract
This paper studies normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically -1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature -1. A relative estimate of Greens function is proved as a tool.
We give a complete description of the global existence and convergence for the Ricci-Yang-Mills flow on $T^k$ bundles over Riemann surfaces. These results equivalently describe solutions to generalized Ricci flow and pluriclosed flow with symmetry.
The main result of this paper shows that, if $g(t)$ is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold $M$ of finite volume, then the Euler characteristic number $chi(M)geq0$. Moreover, $chi(M) eq 0$, there exist a sequence times $t_ktoinfty$, a double sequence of points ${p_{k,l}}_{l=1}^{N}$ and domains ${U_{k,l}}_{l=1}^{N}$ with $p_{k,l}in U_{k,l}$ satisfying the followings: [(i)] $dist_{g(t_k)}(p_{k,l_1},p_{k,l_2})toinfty$ as $ktoinfty$, for any fixed $l_1 eq l_2$; [(ii)] for each $l$, $(U_{k,l},g(t_k),p_{k,l})$ converges in the $C_{loc}^infty$ sense to a complete negative Einstein manifold $(M_{infty,l},g_{infty,l},p_{infty,l})$ when $ktoinfty$; [(iii)] $Vol_{g(t_{k})}(Mbackslashbigcup_{l=1}^{N}U_{k,l})to0$ as $ktoinfty$.
In this note, we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as backwards smooth convergence away from the exceptional curves on compact complex surfaces. The smoothing property for the Chern-Ricci flow is also obtained on compact Hermitian manifolds of dimension n under a mild assumption.
In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In addition, for Ricci harmonic flow, we use the monotonicity of functional $ u_alpha$ to show the connection between finite-time singularity and shrinking Ricci harmonic soliton. At last, we explore the property of ancient solutions for Ricci harmonic flow.
We construct a discrete form of Hamiltons Ricci flow (RF) equations for a d-dimensional piecewise flat simplicial geometry, S. These new algebraic equations are derived using the discrete formulation of Einsteins theory of general relativity known as Regge calculus. A Regge-Ricci flow (RRF) equation is naturally associated to each edge, L, of a simplicial lattice. In defining this equation, we find it convenient to utilize both the simplicial lattice, S, and its circumcentric dual lattice, S*. In particular, the RRF equation associated to L is naturally defined on a d-dimensional hybrid block connecting $ell$ with its (d-1)-dimensional circumcentric dual cell, L*. We show that this equation is expressed as the proportionality between (1) the simplicial Ricci tensor, Rc_L, associated with the edge L in S, and (2) a certain volume weighted average of the fractional rate of change of the edges, lambda in L*, of the circumcentric dual lattice, S*, that are in the dual of L. The inherent orthogonality between elements of S and their duals in S* provide a simple geometric representation of Hamiltons RF equations. In this paper we utilize the well established theories of Regge calculus, or equivalently discrete exterior calculus, to construct these equations. We solve these equations for a few illustrative examples.