| We first present a construction of Einstein metrics which is similar to Thurston's Dehn filling in dimension 3. Motivated by this, we prove a stability result for hyperbolic manifolds with cusps under Ricci flow. In both problems, the main difficulty is coming from a weak stability which arises from so called cusp deformations. We point out two methods how to deal with this weak stability. Then, we show a very strong stability result for general symmetric spaces of noncompact type demonstrating that in certain situations a lack of cusp deformations improves the stability of the underlying space. Moreover, using a new technique developed for this proof, we improve a stability result for real and complex hyperbolic space.;This thesis is a composition of the following three papers of the author: "Construction of Einstein metrics by generalized Dehn filling" ([Bam1]), "Stability of hyperbolic manifolds with cusps under Ricci flow" ([Bam2]), "Stability of symmetric spaces of noncompact type under Ricci flow" ([Bam3]). |
