Length spectrum metric and modified length spectrum metric on Teichmuller spaces

The length spectrum function defines a metric on the reduced Teichmuller space of a Riemann surface which is topologically equivalent, but not metrically equivalent to the Teichmuller metric if the Riemann surface is of finite topological type.;As the first part of this work, in the reduced Teichmuller space of a Riemann surface of finite topological type, we find two points moving towards the boundary of the space along two continuous curves, such that the Teichmuller distance between them approaches infinity while their length spectrum distance approaches zero. Unfortunately, the length spectrum function does not define a metric on the (unreduced) Teichmuller space of a Riemann surface with boundary. In the second part of this work, we introduce a modified length spectrum function that does define a metric on this space. We show that if two points are close with respect to the Teichmuller metric, then they are also close in the modified length spectrum metric. We also show that the converse is not true. Finally, we prove that the (unreduced) Teichmuller space of a Riemann surface of finite topological type with non-empty boundary is not complete under the modified length spectrum metric.