Evolution of Ricci scalar under Finsler Ricci flow

Abstract

Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduced $hh$-curvature on finite time. Next, it is shown that the evolution of Ricci scalar is a parabolic-type equation and if the initial Finsler metric is of positive flag curvature, then the flag curvature and the Ricci scalar remain positive as long as the solution exists. Finally, a lower bound for the Ricci scalar along the Ricci flow is obtained.

17 pages. arXiv admin note: text overlap with arXiv:1508.02893

References in corpus (2)

• On compact Ricci solitons in Finsler geometry
• Hamilton's Ricci Flow on Finsler Spaces