  Let $\pi:\mc{X}\to \mc{T}$ be Teichm\"uller curve over Teichm\"uller space
$\mc{T}$, such that the fiber $\mc{X}_z=\pi^{-1}(z)$ is exactly the Riemann
surface given by the complex structure $z\in \mc{T}$. For a fixed Riemannian
manifold $M$ and a continuous map $u_0: M\to \mc{X}_{z_0}$, let $E(z)$ denote
the energy function of the harmonic map $u(z):M\to \mc{X}_z$ homotopic to
$u_0$, $z\in \mathcal T$. We obtain the first and the second variations of the
energy function $E(z)$, and show that $\log E(z)$ is strictly plurisubharmonic
on Teichm\"uller space, from which we give a new proof on the Steinness of
Teichm\"uller space. We also obtain a precise formula on the second variation
of $E^{1/2}$ if $\dim M=1$. In particular, we get the formula of
Axelsson-Schumacher on the second variation of the geodesic length function. We
give also a simple and corrected proof for the theorem of Yamada, the convexity
of energy function $E(t)$ along Weil-Petersson geodesics. As an application we
show that $E(t)^c$ is also strictly convex for $c>5/6$ and convex for $c=5/6$
along Weil-Petersson geodesics. We also reprove a Kerckhoff's theorem which is
a positive answer to the Nielsen realization problem.

Differential Geometry