Deforming metrics in the direction of their Ricci tensors

@article{DeTurck1983DeformingMI,
  title={Deforming metrics in the direction of their Ricci tensors},
  author={Dennis DeTurck},
  journal={Journal of Differential Geometry},
  year={1983},
  volume={18},
  pages={157-162}
}
  • D. DeTurck
  • Published 1983
  • Mathematics
  • Journal of Differential Geometry
In [4], R. Hamilton has proved that if a compact manifold M of dimension three admits a C Riemannian metric g0 with positive Ricci curvature, then it also admits a metric g with constant positive sectional curvature, and is thus a quotient of the sphere S. In fact, he shows that the original metric can be deformed into the constant-curvature metric by requiring that, for t ≥ 0, x ∈ M and g = g(t, x), 
Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature
The purpose of this paper is to evolve non-smooth Riemannian metric tensors by the dual Ricci-Harmonic map flow. This flow is equivalent (up to a diffeomorphism) to the Ricci flow. One application
Smoothing metrics on closed Riemannian manifolds through the Ricci flow
Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be
A class of Riemannian manifolds that pinch when evolved by Ricci flow
Abstract:The purpose of this paper is to construct a set of Riemannian metrics on a manifold X with the property that will develop a pinching singularity in finite time when evolved by Ricci flow.
Ricci flow and the manifold of Riemannian metrics
R.Hamilton defined Ricci flow as a weak parabolic partial differential equation, in spite of weakness he could prove the existence and uniqueness in the short time, while later DeTurck found a
Ricci Deformation of the Metric on Riemannian Orbifolds
In this note we generalize the Huisken’s (J Diff Geom 21:47–62, 1985) result to Riemannian orbifolds. We show that on any n-dimensional (n ≥ 4) orbifold of positive scalar curvature the metric can be
Four-manifolds with positive isotropic curvature
We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact
Area, Scalar Curvature, and Hyperbolic 3-Manifolds
Let M be a closed hyperbolic 3-manifold that contains a closed immersed totally geodesic surface Σ. Then if g is a Riemannian metric on M with scalar curvature greater than or equal to −6, we show
Infinitesimal Ricci flows of minimal surfaces in the three-dimensional Euclidean space
where Ric(g) is the Ricci tensor of g. The Ricci flow, as a technical tool, was heavily used in works devoted to the proof of the Poincare conjecture, and many results concerning existence and
Stability of the Ricci flow at Ricci-flat metrics
If g is a metric whose Ricci flow g (t) converges, one may ask if the same is true for metrics g that are small perturbations of g. We use maximal regularity theory and center manifold analysis to
Strong uniqueness of the Ricci flow on the Euclidean space in higher dimensions
In this paper, we prove a strong uniqueness theorem for the Ricci flow on the Euclidean space under certain conditions. Let (R, gt)t∈[0,T ], where n ≥ 3, be a Ricci flow solution with g0 = gE , where
...
...

References

SHOWING 1-4 OF 4 REFERENCES
Existence of metrics with prescribed Ricci curvature: Local theory
Foundations of Differentiable Manifolds and Lie Groups
1 Manifolds.- 2 Tensors and Differential Forms.- 3 Lie Groups.- 4 Integration on Manifolds.- 5 Sheaves, Cohomology, and the de Rham Theorem.- 6 The Hodge Theorem.- Supplement to the Bibliography.-
Deforming metrics in the direction of their Ricci tensors
  • J. Differential Geom
  • 1983