MSc Thesis Talk
2.00 pm, Seminar Hall
Collars on Riemann surfaces and its generalization
Chennai Mathematical Institute.
Collars are very useful tools in Riemann surfaces and have many applications. A fundamental property of Riemann surfaces is the existence of collar around a closed geodesic and various versions of the collar theorem are now available. I will first prove the main collar theorem (due to Peter Buser) for closed hyperbolic Riemann surfaces with genus g at least two, by decomposing the surface into $2g-2$ pairs of pants and constructing half collars, which gives a sharp bound for the width of the maximal collar around a simple closed geodesic in terms of its hyperbolic length. Then following the same procedure with some little modification I will prove (due to P Buser again) the collar theorem for variable curvature bounded below by $-1$ and in this case, we also get the same bound for the width of the collar as of the previous case. Then I will talk about the generalized collar theorem (due to Ara Basmajian) in which case the underlying space is the same as of the first case but the closed geodesic need not be simple here and 'collar' is replaced by something called 'stable neighborhood'. Assuming some results I will give a sketch of the proof for the last one.