Journal of the
Korean Mathematical Society JKMS

Here, we calculate the evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow. We prove that Finsler Ricci flow preserves positivity of the reduced $hh$-curvature on finite time. Next, it is shown that evolution of Ricci scalar is a parabolic-type equation and moreover if the initial Finsler metric is of positive flag curvature, then the flag curvature, as well as the Ricci scalar, remain positive as long as the solution exists. Finally, we present a lower bound for Ricci scalar along Ricci flow.

Keywords: Finsler, evolution, Ricci flow, Ricci scalar, maximum principle, parabolic equation

MSC numbers: 53C60, 53C44