Eigenvalues and Entropy of a Hitchin representation

We show that the critical exponent of a representation in the Hitchin component of $PSL(d,\mathbb{R})$ is bounded above, the least upper bound being attained only in the Fuchsian locus. This provides a rigid inequality for the area of a minimal surface on $\rho\backslash X,$ where $X$ is the symmetric space of $PSL(d,\mathbb{R}).$ The proof relies in a construction useful to prove a regularity statement: if the Frenet equivariant curve of $\rho$ is smooth, then $\rho$ is Fuchsian.