Problems around 3–manifolds

This is a personal view of some problems on minimal surfaces, Ricci flow, polyhedral geometric structures, Haken 4–manifolds, contact structures and Heegaard splittings, singular incompressible surfaces after the Hamilton–Perelman revolution. We give sets of problems based on the following themes; Minimal surfaces and hyperbolic geometry of 3–manifolds. In particular, how do minimal surfaces give information about the geometry of hyperbolic 3–manifolds and conversely how does the geometry affect the types of minimal surfaces which can occur? Ricci flow. Here it would be good to be able to visualize Ricci flow and understand more about where and how singularities form. Also this leads into the next topic, via the possibility of a combinatorial version of Ricci flow. Combinatorial geometric structures. Various proposals have been made for combinatorial versions of differential geometry in dimension 3. However having a good model of sectional or Ricci curvature is a difficult challenge. Haken 4–manifolds. These are a class of 4–manifolds which admit hierarchies, in the same sense as Haken 3–manifolds. More information is needed to determine how abundant they are. Contact structures and Heegaard splittings. Contact structures are closely related to open book decompositions, which can be viewed as special classes of Heegaard splittings. Singular incompressible surfaces. A well-known and important conjecture is that all (closed or complete finite volume) hyperbolic 3–manifolds admit immersed or embedded closed incompressible surfaces. We give some variants of this problem.