On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons

In this article, we investigate the geometry of 4 4 -dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a 4 4 -dimensional compact gradient Ricci soliton must satisfy the classical Hitchin-Thorpe inequality. In addition, some volume estimates are also obtained.