Rigidity of the critical point equation

✍ Scribed by Seungsu Hwang; Jeongwook Chang; Gabjin Yun

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 119 KB
• Volume: 283
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200710037

✦ Synopsis

Abstract

On a compact n ‐dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation ([5], p. 3222). In 1987 Besse proposed a conjecture in his book [1], p. 128, that a solution of the critical point equation is Einstein (Conjecture A, hereafter). Since then, number of mathematicians have contributed for the proof of Conjecture A and obtained many geometric consequences as its partial proofs. However, none has given its complete proof yet.

The purpose of the present paper is to prove Theorem 1, stating that a compact 3‐dimensional manifold M is isometric to the round 3‐sphere S^3^ if ker s^′*^~g~ ≠ 0 and its second homology vanishes. Note that this theorem implies that M is Einstein and hence that Conjecture A holds on a 3‐dimensional compact manifold under certain topological conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)