Let $(M, g, f)$ be a $4$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+ abla^2f=lambda g$, where $lambda$ is a positive real number. We prove that if $M$ has constant scalar curvature $S=2lambda$, it must be
a quotient of $mathbb{S}^2times mathbb{R}^2$. Together with the known results, this implies that a $4$-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton $Bbb{R}^4$, $Bbb{S}^{2}timesBbb{R}^{2}$ or $Bbb{S}^{3}timesBbb{R}$.
In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part o
f the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton $Bbb{R}^4,$ or $Bbb{S}^{3}timesBbb{R}$, or $Bbb{S}^{2}timesBbb{R}^{2}.$ In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.
In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different meth
od, we generalize maximum principles of Omori and Yau to a viscosity version.