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Rigidity of the critical point equation
Author(s) -
Hwang Seungsu,
Chang Jeongwook,
Yun Gabjin
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200710037
Subject(s) - mathematics , conjecture , mathematical proof , scalar curvature , manifold (fluid mechanics) , rigidity (electromagnetism) , einstein , pure mathematics , critical point (mathematics) , curvature , mathematical analysis , mathematical physics , geometry , physics , mechanical engineering , quantum mechanics , engineering
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)