Reducible handle additions to weakly reducible Heegaard splittings (Q1637107)

If \(M\) is a hyperbolic \(3\)-manifold with a torus boundary component, Thurston showed that only finitely many of the possible Dehn fillings yield non-hyperbolic \(3\)-manifolds. If \(M\) is a hyperbolic \(3\)-manifold with boundary of genus greater than one, there are known examples such that infinitely many distinct handle additions yield non-hyperbolic \(3\)-manifolds. However, \textit{M. Scharlemann} and \textit{Y.-Q. Wu} [J. Aust. Math. Soc., Ser. A 55, No. 1, 72--89 (1993; Zbl 0802.57005)] showed that the set of basic degenerating curves is finite and bounded by a universal constant only depending on the genus of the boundary. The first two authors and \textit{F. Lei} [Proc. Am. Math. Soc. 144, No. 1, 423--434 (2016; Zbl 1327.57022) gave the diameter of the set of degenerating curves by using subsurface projections if \(M=V\cup_{S} W\) is a strongly irreducible Heegaard splitting. In the paper under review, the authors consider a weakly reducible Heegaard splitting and give the diameter of the set of degenerating curves by using subsurface projections. Also, the authors construct a keen weakly reducible Heegaard splitting.