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As indicated by the third author in [19], there is a gap in the previousversion of this paper by the first two authors [5]. We provide in this versionan argument to fix the aforementioned gap. The main proposition, whose proofuses Perelman's techniques, is implied by Ding [9] and is covered by [19]. Ourapproach, however, is different from theirs. In addition, we prove a necessaryand sufficient condition for a three-dimensional $\kappa$-solution to form aforward singularity. We hope that this condition is helpful in theclassification of all three-dimensional $\kappa$-solutions. Up to now, the onlymain progress on such a classification, as conjectured by Perelman, is byBrendle [2].

Three-dimensional noncompact $\kappa$-solutions that are Type I forward and backward