Compact 8-manifolds with holonomy Spin(7)

Taylor, Christine

doi:10.1007/s002220050039

Cited by 130 publications

(304 citation statements)

References 18 publications

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“…Similar technique was later used by Joyce [30,31] in 1996 to construct seven dimensional manifolds with holonomy group equal to G 2 and eight dimensional manifolds with holonomy group equal to Spin (7).…”

Section: G/h G/kmentioning

confidence: 95%

A survey of geometric structure in geometric analysis

Yau

2011

Surveys in Differential Geometry

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The subject of geometric analysis evolves according to our understanding of geometry and analysis. However, one should say that ideas of algebraic geometry and representation theory have been extremely powerful in both global and local geometry.In fact, the spectacular idea of using geometry to understand Diophantine problem has already widened our concept of space. The desire to find suitable geometry to accommodate unified field theory in physics would certainly drastically change the scope of geometry in the near future.In the following lectures, we shall focus on an important branch of geometric analysis: the construction of geometric structures over a given topological space.

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Section: G/h G/kmentioning

confidence: 95%

A survey of geometric structure in geometric analysis

Yau

2011

Surveys in Differential Geometry

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“…There are many known examples of Joyce spin(7) orbifolds [14]. For simplicity, I will take one orbifold for example in which we have choices in desingularizing T …”

Section: Joyce's Construction Of Spin(7) Manifoldsmentioning

confidence: 99%

A Note on Mirror Symmetry for Manifolds with Spin(7) Holonomy

Chuang¹

2004

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“…This method has been used to construct solutions of many nonlinear elliptic equations arising in differential geometry (see e.g. [2], [3], [6], [8], [19], [23], [24], [25], [26], [28], [27], [30], [31], [36], [39], [40]). …”

Section: Introductionmentioning

confidence: 99%

“…This construction also shares some common features with Joyce's construction of manifolds with exceptional holonomy (cf. [23]). …”

Section: Introductionmentioning

confidence: 99%

Gluing Eguchi‐Hanson Metrics and a Question of Page

Brendle¹,

Kapouleas²

2016

Comm Pure Appl Math

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Abstract. In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat Kähler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with 16 orbifold points, and resolves the orbifold singularities by gluing in 16 small Eguchi-Hanson manifolds which all have the same orientation. This construction was carried out rigorously by Topiwala, LeBrunSinger, and Donaldson.In 1981, Page asked whether the above construction can be modified by reversing the orientations of some of the Eguchi-Hanson manifolds. This is a subtle question: if successful, this construction would produce Einstein metrics which are neither Kähler nor self-dual.In this paper, we focus on a configuration of maximal symmetry involving 8 small Eguchi-Hanson manifolds of each orientation which are arranged according to a chessboard pattern. By analyzing the interactions between Eguchi-Hanson manifolds with opposite orientation, we identify a non-vanishing obstruction to the gluing problem, thereby destroying any hope of producing a metric of zero Ricci curvature in this way. Using this obstruction, we are able to understand the dynamics of such metrics under Ricci flow as long as the Eguchi-Hanson manifolds remain small. In particular, for the configuration described above, we obtain an ancient solution to the Ricci flow with the property that the maximum of the Riemann curvature tensor blows up at a rate of (−t) 1 2 , while the maximum of the Ricci curvature converges to 0.

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Compact 8-manifolds with holonomy Spin(7)

Cited by 130 publications

References 18 publications

A survey of geometric structure in geometric analysis

A survey of geometric structure in geometric analysis

A Note on Mirror Symmetry for Manifolds with Spin(7) Holonomy

Gluing Eguchi‐Hanson Metrics and a Question of Page

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