Erratum to “A gap theorem for hypersurfaces with constant scalar curvature one” | Zendy

Hilário Alencar | Zendy; Walcy Santos | Zendy; Manfredo do Carmo | Zendy

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Erratum to “A gap theorem for hypersurfaces with constant scalar curvature one”

Author(s) -

Hilário Alencar,

Walcy Santos,

Manfredo do Carmo

Publication year - 2006

Publication title -

commentarii mathematici helvetici

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.603

H-Index - 46

eISSN - 1420-8946

pISSN - 0010-2571

DOI - 10.4171/cmh/44

Subject(s) - mathematics , orthonormal basis , lemma (botany) , mathematical analysis , immersion (mathematics) , pure mathematics , gap theorem , constant (computer programming) , scalar curvature , principal curvature , curvature , mathematical physics , compactness theorem , geometry , fundamental theorem , physics , fixed point theorem , computer science , programming language , ecology , poaceae , quantum mechanics , biology

Here Sr , r = 1, . . . , n, is the r th-symmetric function of the principal curvatures of the immersion and L1 is the linearized operator corresponding to the equation of hypersurfaces of Sn+1 with S2 ≡ 0. The proof of Lemma 4.1 presented in [AdCS] is incorrect. We had set N = ∑n+2 i=1 niei , where {e1, . . . , en+2} is an orthonormal basis of Rn+2 and N is the unit normal vector to the immersion of M → Sn+1 ⊂ Rn+2. By assuming the index of I is greater than one and noticing that I (ni, ni) ≤ 0, we concluded that all ni but one were zero. This is not true. Replace the proof of the lemma by the following.

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