Geometric and topological rigidity theorem for 3-manifolds

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The researcher has studied thoroughly geometric and topological rigidity theorems for 3-manifolds. In particular, he found out that the existence of least area planes properly embedded in the universal coverings in the proof of topological rigidity theorems. Let M be a closed hyperbolic 3-manifold and p:H^3→M the universal covering. Here, we suppose that M has a Riemannian metric which is not necessarily hyperbolic. The metric r on H^3 induced from that on M is called a co-compact metric. D.Gabai conjectured that "any simple smooth curve in the boundary S^2_∞ of H^3 spans a properly embedded r-least area plane in H^3"(J.Amer.Math.Soc.10(1997)). Throughout this project, the researcher proved that the conjecture is true. Moreover, he proved that the result holds when π_1(M) is Gromov-hyperbolic even if M is not a hyperbolic 3-manifold. That is, it was shown that, for the universal converging M^^〜 of the manifold M, any Jordan curve in ∂M^^〜 bounds a properly embedded r-least area plane in M.Furthermore, the researcher solved the question "What kinds of topological types do geometric limits of quasi-Fuchsian groups have ?" completely. Precisely, Σis a closed orientable surface of genus>1, and {p_n} is an algebraically convergent sequence of quasi-Fuchsian representations ρ_n:π_1(Σ)→PSL_2(C). Suppose that the sequence {Γ_n} consisting of the quasi-Fuchsian groups Γ_n=ρ_n(π_1(Σ)) converges geometrically to a Kleinian group G. Then, the researcher proved that there exists a closed set Χ in Σ×[0,1] called a crevasse so that H^3/G is homeomorphic to Σ×[0,1]-Χ. Conversely, it was also proved that, for any crevasse Χ in Σ×[0,1], there exists a geometric, limits G of quasi-Fuchsian groups such that H^3/G is homeomorphic to Σ×[0,1]-Χ.