  A basic feature of Teichm\"uller theory of Riemann surfaces is the interplay
of two dimensional hyperbolic geometry, the behavior of geodesic-length
functions and Weil-Petersson geometry. Let $\mathcal{T}_g$ $(g\geq 2)$ be the
Teichm\"uller space of closed Riemann surfaces of genus $g$. Our goal in this
paper is to study the gradients of geodesic-length functions along systolic
curves. We show that their $L^p$ $(1\leq p \leq \infty)$-norms at every
hyperbolic surface $X\in \mathcal{T}_g$ are uniformly comparable to
$\ell_{sys}(X)^{\frac{1}{p}}$ where $\ell_{sys}(X)$ is the systole of $X$. As
an application, we show that the minimal Weil-Petersson holomorphic sectional
curvature at every hyperbolic surface $X\in \mathcal{T}_g$ is bounded above by
a uniform negative constant independent of $g$, which negatively answers a
question of M. Mirzakhani. Some other applications to the geometry of
$\mathcal{T}_g$ will also be discussed.
