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Deformations of the hemisphere that increase scalar curvature. (English) Zbl 1227.53048

The authors disprove the Min-Oo conjecture [M. Min-Oo, “Scalar curvature rigidity of certain symmetric spaces”, CRM Proc. Lect. Notes 15, 127–136 (1998; Zbl 0911.53032)], which claims that a Riemannian metric \(g\) on the hemisphere \(S_+^n\), \(n \geq 3\), which coincides with the standard metric on the boundary, and which satisfies the following two properties:
1. the boundary is totally geodesic with respect to \(g\),
2. the scalar curvature of \(g\) is at least \(n(n-1)\),
is the standard spherical metric.
This conjecture was supported by a result by Miao (2002) about an analogous characterization of the Euclidean metric and by the generalization of a theorem of Miao by Y. Shi and L. F. Tam (2002).

MSC:

53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 0911.53032
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References:

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