On the Instability of a Minimal Surface in a 4-Manifold Whose Curvature Lies in the Interval (1/4, 1]

Kawai, Shigeo

doi:10.2977/prims/1195183296

Cited by 2 publications

(4 citation statements)

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“…In order to prove Theorem C, we shall recall some useful facts in Refs. [19,24], and we will also make some additional observation.…”

Section: Instability Of Some Minimal 2-spheresmentioning

confidence: 97%

“…Therefore, we are going to pick up some normal cross sections, which are 'half-parallel' or 'pseudoholomorphic' [19] (or see Ref. [14]).…”

Section: Twisted Property Of Normal Bundlementioning

confidence: 99%

“…Let {e 3 , e 4 } be any positively oriented orthonormal basis of N p (S), we let Je 3 = e 4 and Je 4 = −e 3 . It is easy to check [19] …”

Section: Twisted Property Of Normal Bundlementioning

confidence: 99%

“…We may always assume that c 1 (ν) 0 (see Ref. [19]). When S is an immersed minimal sphere, the existence of a holomorphic normal cross section is given by the RiemannRoch theorem.…”

Section: It Follows Form Definition 31 (3) and (37) Thatmentioning

confidence: 99%

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Certain 4-manifolds with non-negative sectional curvature

Cao

2008

Front. Math. China

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Abstract.In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W + is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S 2 × S 2 with both (i) K > 0 and (ii)We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: "If a simply-connected, closed 4-manifold M 4 admits a metric g of non-negative curvature operator, then M 4 is one of S 4 , CP 2 and S 2 × S 2 ".Our method is different from Hamilton's and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.

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“…In order to prove Theorem C, we shall recall some useful facts in Refs. [19,24], and we will also make some additional observation.…”

Section: Instability Of Some Minimal 2-spheresmentioning

confidence: 97%

“…Therefore, we are going to pick up some normal cross sections, which are 'half-parallel' or 'pseudoholomorphic' [19] (or see Ref. [14]).…”

Section: Twisted Property Of Normal Bundlementioning

confidence: 99%

“…Let {e 3 , e 4 } be any positively oriented orthonormal basis of N p (S), we let Je 3 = e 4 and Je 4 = −e 3 . It is easy to check [19] …”

Section: Twisted Property Of Normal Bundlementioning

confidence: 99%

“…We may always assume that c 1 (ν) 0 (see Ref. [19]). When S is an immersed minimal sphere, the existence of a holomorphic normal cross section is given by the RiemannRoch theorem.…”

Section: It Follows Form Definition 31 (3) and (37) Thatmentioning

confidence: 99%

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