Ricci flow does not preserve positive sectional curvature in dimension four

Bettiol, Renato G.; Krishnan, Anusha M.

doi:10.1007/s00526-022-02335-z

Cited by 4 publications

(3 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There exist examples in infinitely many dimensions where positive Ricci curvature is not preserved [AN16], as well as examples in dimensions 4, 6, 12, 24 where positive sectional curvature is not preserved [BK23,BW07,CW15]. In this context, the example in dimension 12 is specially relevant to us.…”

Section: Corollary Bmentioning

confidence: 99%

Positive intermediate curvatures and Ricci flow

González-Álvaro¹,

Zarei²

2023

Preprint

View full text Add to dashboard Cite

We show that, for any n ≥ 2, there exists a homogeneous space of dimension d = 8n − 4 with metrics of Ric d 2 −5 > 0 if n = 3 and Ric 6 > 0 if n = 3 which evolve under the Ricci flow to metrics whose Ricci tensor is not (d − 4)-positive. Consequently, Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. This extends a theorem of Böhm and Wilking in the case of n = 2.

show abstract

Section: Corollary Bmentioning

confidence: 99%

Positive intermediate curvatures and Ricci flow

González-Álvaro¹,

Zarei²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…For d = 1, 2, 3 let us take t 0 ∈] 5 16 , 1 3 [ and g = (t 0 , t 0 , 1 − 2t 0 ). Observe that such a metric has positive Ric d > 0 for d = 1, 2, accordingly to Proposition 3.5.…”

Section: The Projected Homogeneous Ricci Flow In Su(3)/tmentioning

confidence: 99%

“…Particularly to the sub-field of positive curvatures, the inaugural work of Böhm and Wilking [7] establishes the non-maintenance of positive Ricci curvature on a certain 12-dimensional homogeneous space under the Ricci flow. Several works appear analyzing this and related phenomena, [5,2,18,24].…”

Section: Introductionmentioning

confidence: 99%

On the dynamics of positively curved metrics on SU(3)/T 2 under the homogeneous Ricci flow

Cavenaghi,

Grama,

Martins

2024

View full text Add to dashboard Cite

In this note, we show that the classical Wallach manifold SU(3)/T 2admits metrics of positive intermediate Ricci curvature (Ric d > 0) for d = 1, 2, 3, 4, 5 that lose these properties under the homogeneous Ricci flow for d = 1, 2, 3, 5. We make the same analyses to the family of Riemannian flag manifolds SU(m + 2p)/S(U(m) × U(p) × U(p)), concluding similar results. These explicitly verify some claims expected to be true among experts (see [7]) for positive Ricci curvature and intermediate positive Ricci curvature. Our technique is only possible due to the global behavior understanding of the homogeneous Ricci flow for invariant metrics on these manifolds.

show abstract