BOL 8(3(3. BRAS. MAT., VOL. 18 N ~ 2 (1987), 83-94 83 CORRECTION OF COMPLETE HYPERSURFACES IN THE SPACE FORM WITH THREE PRINCIPAL CURVATURES REIKO MIYAOKA In the proofs of following two theorems in [2], some mistakes were found out by Professor Pawel Walczak and Professor Fabiano Brito. Theorem I. Let M be a complete hypersurface in ~(o) with constant.mean curvature, where o > O. If M has three non-simple principai curvatures, then M is isoparametric and c > 0. Theorem II. Let M be a complete minimal hypersurface in S n+l with three principal curvatures. If n > 4, then M is either isoparametric, of type (BI), or of type (B2) in Ill. Here we correct both proofs following the original method. The essential miss is in the proof of case (iii) appeared in p. 352 of [2]. By the way, we give another proof for Theorem I in compact case using the results in ~]. This is equivalent to prove that Theorem III. If a compact embedded Dupin hypersurface M in M(e) (e ~ 0) with three principal curvatures has a constant mean curvature, then M is isoparametric and c > 0, In this paper, we fo'llow the notation in ~], Recall that ~{c) : E n+l or S n+l : sn+l(1) according to c = 0 or c : I. Principal curvatures of M are ~, ~, v satisfying /'ttl~. + m2]J -I- m3%) = nh, 'Recebido em 13/05/87. Revised copy 27/10/87.