New gap results on the 4†dimensional sphere

Ezequiel Barbosa, Allan Freitas and Antonio Rosivaldo Gonçalves

Mathematische Nachrichten, 2017, vol. 290, issue 17-18, 2755-2758

Abstract: A result showed by M. Gursky in ensures that any metric g on the 4†dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥π34 is isometric to the round metric. In this note, we prove that there exists a universal number i0 such that any metric g on the 4†dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥π34−i0 is isometric to the round metric. Moreover, there exists a universal ε0>0 such that any metric g on the 4†dimensional sphere S4 with nonnegative sectional curvature, Ricg=3g and 89π2−ε0≤Volg(S4) is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in .

Date: 2017
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