Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature

@article{Bettiol2019FourdimensionalCO,
  title={Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature},
  author={Renato G. Bettiol and Anusha M. Krishnan},
  journal={Communications in Analysis and Geometry},
  year={2019}
}
We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow. 

Figures and Tables from this paper

Ricci flow does not preserve positive sectional curvature in dimension four
We find examples of cohomogeneity one metrics on S and CP 2 with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of
Convergence of Ricci flow solutions to Taub-NUT
Abstract We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric g 0 on with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If g 0 has bounded
On the stability of harmonic maps under the homogeneous Ricci flow
Abstract In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow.We provide examples where the stability (non-stability) is preserved under
Ricci Flow On Cohomogeneity One Manifolds
RICCI FLOW ON COHOMOGENEITY ONE MANIFOLDS Anusha Mangala Krishnan Wolfgang Ziller In the first part of this thesis, in joint work with Renato Bettiol, we show that the geometric property of
Diagonalizing the Ricci Tensor
We show that a basis of a semisimple Lie algebra for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being ``nice''. Namely,
Convergence of Ricci flow solutions to Taub-NUT
We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric $g_{0}$ on $\mathbb{R}^{4}$ with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If
The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds
Let $$G{/}H$$G/H be a compact homogeneous space, and let $$\hat{g}_0$$g^0 and $$\hat{g}_1$$g^1 be G-invariant Riemannian metrics on $$G/H$$G/H. We consider the problem of finding a G-invariant
Ricci flow of warped Berger metrics on $${\mathbb {R}}^{4}$$
We study asymptotically flat SU(2)-cohomogeneity 1 solutions of Ricci flow on $\mathbb{R}^{4}$ whose restrictions to any Euclidean hypersphere are left-invariant Berger spheres. We show that these
...
...

References

SHOWING 1-10 OF 35 REFERENCES
On the Geometry of Cohomogeneity One Manifolds with Positive Curvature
We discuss manifolds with positive sectimal curvature on which a group acts isometrically with one dimensional quotient. A number of the known examples have this property, but some potential families
Manifolds with 1/4-pinched curvature are space forms
Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also
Immortal homogeneous Ricci flows
We show that for an immortal homogeneous Ricci flow solution any sequence of parabolic blow-downs subconverges to a homogeneous expanding Ricci soliton. This is established by constructing a new
On the blow-up of four dimensional Ricci flow singularities
In this paper we prove a conjecture by Feldman-Ilmanen-Knopf in \cite{FIK} that the gradient shrinking soliton metric they constructed on the tautological line bundle over $\CP^1$ is the uniform
On Ricci solitons of cohomogeneity one
We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansätze of cohomogeneity one to produce new explicit examples of complete Kähler Ricci solitons of expanding,
Lie Groups and Geometric Aspects of Isometric Actions
1: Basic results on Lie groups.- 2: Lie groups with bi-invariant metrics.- 3: Proper and isometric acions.- 4: Adjoint and conjugation actions.- 5: Polar foliations.- 6: Low cohomogeneity actions and
Ricci flow and nonnegativity of sectional curvature
In this paper, we extend the general maximum principle in (NT3) to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As
Non-negative Ricci curvature on closed manifolds under Ricci flow
In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K}
Backwards uniqueness of the Ricci flow
In this paper, we prove a unique continuation or ``backwards-uniqueness'' theorem for solutions to the Ricci flow. A particular consequence is that the isometry group of a solution cannot expand
Isotropic Curvature and the Ricci Flow
In this paper, we study the Ricci flow on higher dimensional compact manifolds. We prove that nonnegative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to
...
...