# New K\"ahler metric on quasifuchsian space and its curvature properties

## TL;DR

This paper introduces a new Kähler metric on quasifuchsian space extending the Weil-Petersson metric, and analyzes its curvature properties, revealing negativity along certain directions, thus advancing understanding of the geometric structure of $QF(S)$.

## Contribution

The authors construct a novel mapping class group invariant Kähler metric on quasifuchsian space that extends the Weil-Petersson metric and analyze its curvature properties.

## Key findings

- The new metric extends the Weil-Petersson metric on Teichmüller space.
- Curvature calculations show negativity along tautological directions.
- The metric is invariant under the mapping class group.

## Abstract

Let $QF(S)$ be the quasifuchsian space of a closed surface $S$ of genus $g\geq 2$. We construct a new mapping class group invariant K\"ahler metric on $QF(S)$. It is an extension of the Weil-Petersson metric onthe Teichm\"uller space $\mathcal T(S)\subset QF(S)$. We also calculate its curvature and prove some negativity for the curvature along the tautological directions.