Chennai Mathematical Institute


3.30 pm -- 4.30 pm, Lecture Hall 6
MSc Student Talks
Collars on Riemann Surfaces

Satyajit Maity
Chennai Mathematical Institute.


Let $S$ be a closed Riemann Surface (with genus at least two) with a hyperbolic metric and $ \gamma $ be a simple closed geodesic in $S$. Then a very basic and fundamental property of Riemann Surfaces is that for small values of $d$ the family of geodesic segments of length $2d$ perpendicular to and centered on $ \gamma $ sweeps out a region (which is topologically homeomorphic to a cylinder basically) called a "collar" around $ \gamma $ of width $d$ which is defined to be the set of all points in $S$ whose hyperbolic distance from $ \gamma $ is less than $d$. So now one could ask a natural question : how much one can enlarge the collar? Or, how big collar one could get around a simple closed geodesic of a given length? In 1973, Linda Keen first answered to this question and later in 1976, J P Matelski extended the result to the case of an arbitrary Riemann surfaces. Their proofs are Fuchsian group theoretic and I don't know whether their answers are best possible or not. But in 1979, Burton Randol treated the question (with the original assumption on $S$) in an elementary geometric way and gave a best possible answer. In my talk, I am going to present the Randol's proof where "how big collar" is going to be answered by its hyperbolic area. More precisely, I will show that for a given simple closed geodesic $ \gamma $ in $S$ of length, say, $L$ one can get collar $ C_\gamma $ around $ \gamma $ whose hyperbolic area is greater than or equal to $ 2L \operatorname{csch(L/2)} $. And it is the best possible estimation in the sense that there are examples for which this lower bound is the maximum possible area of a collar one can get for the given length $L$ of $\gamma$.

As prerequisites familiarity with differential geometry, Riemannian geometry and elementary hyperbolic geometry will be helpful.