Positive isotropic curvature and self-duality in dimension 4

Richard, Thomas; Seshadri, Harish

doi:10.1007/s00229-015-0790-2

Cited by 8 publications

(6 citation statements)

References 26 publications

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“…For Einstein manifolds with PIC, Brendle [4] proved that they must be isometric to the round sphere S n , up to scaling. Later, in dimension n = 4, this result was extended by Richard-Seshadri [46], in which they showed that a compact oriented Einstein 4-manifold with half PIC is isometric to S 4 or CP 2 . On the other hand, for gradient shrinking Ricci solitons, it was proved recently by Li-Ni-Wang [31] that any 4-dimensional complete gradient shrinking Ricci soliton with PIC is a finite quotient of either S 4 or S 3 × R.…”

Section: Introductionmentioning

confidence: 91%

Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature

Cao¹,

Xie²

2022

Preprint

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In this paper, we investigate the geometry of 4-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature (half PIC) or half nonnegative isotropic curvature. Our first main result is a certain quadratic curvature lower bound estimate for such gradient Ricci shrinkers. As a consequence, we obtain a new proof of the classification result first shown by Li-Ni-Wang [31] for gradient shrinking Kähler-Ricci solitons of complex dimension two with nonnegative isotropic curvature. Moreover, we classify 4-dimensioinal complete gradient shrinking Ricci solitons with half PIC under an additional assumption that their Ricci tensor has an eigenvalue with multiplicity 3.

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Section: Introductionmentioning

confidence: 91%

Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature

Cao¹,

Xie²

2022

Preprint

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“…In [RS13b], H. Seshadri and the author used the results from [RS13a] to prove that the cones C IC + and C IC − enjoy some kind of maximality among oriented Ricci flow invariant nonnegative curvature cones in dimension 4.…”

Section: The Case Of Dimensionmentioning

confidence: 99%

Curvature cones and the Ricci flow.

Richard¹

2014

Séminaire De Théorie Spectrale Et Géométrie

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This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points.First we describe the known examples of preserved curvature conditions and how they have been used to derive geometric results, in particular sphere theorems.We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters.The Ricci flow is the following evolution equation:where (g(t)) t∈[0,T ) is a one-parameter family of smooth Riemannian metrics on a fixed manifold M , and g 0 is a given smooth Riemannian metric on M . It was introduced by R. Hamilton in 1982 ([Ham82]), where it was used to study the topology of compact 3-manifolds with positive Ricci curvature. Analytically, the Ricci flow is a degenerate parabolic system. Existence and uniqueness for the Cauchy problem (1) . Since Hamilton's work, the Ricci flow has been used to solve various geometric problems. We refer to the previously cited books for examples. Here we will just briefly mention two of the biggest geometric achievements of Ricci flow:• The proof of Thurston's Geometrization conjecture for 3-manifolds by G. The objects we will be dealing with in this survey have little to do with Perelman's work, but were pivotal in the proof of the differentiable sphere theorem.A priori estimates are among the most basic tools in the study of PDEs, one can divide them into two classes: integral estimates and pointwise estimates. Pointwise estimates often come from suitable maximum principles and are thus more often encountered in the realm of parabolic or elliptic equations. The Ricci flow being a geometric parabolic PDE, it is tempting to look for geometrically meaningful pointwise estimates for Ricci flows.If one looks at Hamilton's foundational work, one sees that after having proven short time existence and derived some variation formulas for the Ricci flow, Hamilton proves the following result:Proposition 0.1. Let (M 3 , g 0 ) be a compact 3-manifold with nonnegative Ricci curvature, then the solution g(t) to (1) with initial condition g 0 satisfies Ric g(t) ≥ 0 for all t ≥ 0. This is the kind of geometric pointwise estimate we will be concerned with. More precisely, we will try to gather what is known about various nonnegativity properties of the curvature of Riemannian manifold (M, g 0 ) which remain valid for solutions g(t) of the Ricci flow starting at g 0 .Let us now describe the contents of this paper. Section 1 sets the scene by introducing an abstract framework which allows us to consider nonnegativity conditions on the curvature as convex cones inside some vector space satisfying some invariance properties: the so-called curvature cones. We then describe how Hamilton's maximum principle characterizes which curvature cones lead to a nonnegativity condition on the curvature which is preserved under the action of the Ricci flow. In section 2 we review the...

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“…Richard and Seshadri [14], Fine, Krasnov, and Panov [5], and the author [19] proved that if the metric has half nonnegative isotropic curvature, then it is either half conformally flat or Kähler. LeBrun [8] proved that if W + (ω, ω) > 0 for some ω ∈ H 2 + (M ), then the metric is Hermitian.…”

Section: Introductionmentioning

confidence: 99%

Einstein four-manifolds with self-dual Weyl curvature of nonnegative determinant

2019

Preprint

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Positive isotropic curvature and self-duality in dimension 4

Cited by 8 publications

References 26 publications

Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature

Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature

Curvature cones and the Ricci flow.

Einstein four-manifolds with self-dual Weyl curvature of nonnegative determinant

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