The Ricci flow on the 2-sphere

@article{Chow1991TheRF,
  title={The Ricci flow on the 2-sphere},
  author={Bennett Chow},
  journal={Journal of Differential Geometry},
  year={1991},
  volume={33},
  pages={325-334}
}
  • B. Chow
  • Published 1991
  • Mathematics
  • Journal of Differential Geometry
The classical uniformization theorem, interpreted differential geomet-rically, states that any Riemannian metric on a 2-dimensional surface ispointwise conformal to a constant curvature metric. Thus one can con-sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met-ric. The primary interest in this question is not so much to give a newproof of the uniformization theorem, but rather to understand… 
Ricci Flow and the Sphere Theorem
In 1982, R. Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds. This evolution equation is known as the Ricci flow, and
SINGULARITIES OF THE RICCI FLOW ON 3-MANIFOLDS
We present an overview of the singularity formation of the Ricci flow on 3-manifolds. The article, is the written version of the talks I gave at the BIRS Workshop on Geometric Flows in Mathematics
Uniformization of conformally finite Riemann surfaces by the Ricci flow
In this paper we give a new proof of the uniformization of conformally finite Riemann surface of negative Euler characteristic by the Ricci flow. Specifically, we will consider the normalized Ricci
Higher order curvature flows on surfaces
We consider a sixth and an eighth order conformal flow on Riemannian surfaces, which arise as gradient flows for the Calabi energy with respect to a higher order metric. Motivated by a work of Struwe
An Introduction to Ricci Flow for Two-Dimensional Manifolds
The study of differentiable manifolds is a deep an extensive area of mathematics. A technique such as the study of the Ricci flow turns out to be a very useful tool in this regard. This flow is an
Scalar Curvature, Conformal Geometry, and the Ricci Flow with Surgery
In this note we will review recent results concerning two geometric problems associated to the scalar curvature. In the first part we will review the solution to Schoen’s conjecture about the
Ricci flow on surfaces along the standard lightcone in the $3+1$-Minkowski spacetime
Identifying any conformally round metric on the 2-sphere with a unique cross section on the standard lightcone in the 3+1-Minkowski spacetime, we gain a new perspective on 2d-Ricci flow on
Normalized Ricci Flow on Riemann Surfaces and Determinant of Laplacian
In this letter, we give a simple proof of the fact that the determinant of Laplace operator in a smooth metric over compact Riemann surfaces of an arbitrary genus g monotonously grows under the
(KÄHLER-)RICCI FLOW ON (KÄHLER) MANIFOLDS
One of the most interesting questions in Riemannian geometry is that of deciding whether a manifold admits curvatures of certain kinds. More specifically, one might want to know whether some given
...
...

References

SHOWING 1-5 OF 5 REFERENCES
On the parabolic kernel of the Schrödinger operator
Etude des equations paraboliques du type (Δ−q/x,t)−∂/∂t)u(x,t)=0 sur une variete riemannienne generale. Introduction. Estimations de gradients. Inegalites de Harnack. Majorations et minorations des
Comparison theorems in Riemannian geometry
Basic concepts and results Toponogov's theorem Homogeneous spaces Morse theory Closed geodesics and the cut locus The sphere theorem and its generalizations The differentiable sphere theorem Complete
Anti Self‐Dual Yang‐Mills Connections Over Complex Algebraic Surfaces and Stable Vector Bundles
On presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels. Soit une surface algebrique projective X qui a un plongement donne X≤CP N et soit ω
On the positivity of the effective action in a theory of random surfaces
AbstractIt is shown that the functional $$S[\eta ] = \frac{1}{{24\pi }}\int {\left( {\frac{1}{2}\left| {\nabla \eta } \right|^2 + 2\eta } \right)d\mu _0 }$$ , defined onC∞ functions on the