I will summarize the work I have done both myself and with my coauthors during my Ph.D. studies. This includes the estimates of moduli of continuity for viscosity solutions in domains in Euclidean spaces and on manifolds, asymptotic behavior of nonparametric hypersurfaces moving by powers of Gauss Curvature, and the classification of four-dimensional shrinking gradient Ricci solitons with positive isotropic curvature.