On the microscopic spacetime convexity principle of fully nonlinear parabolic equations II: Spacetime quasiconcave solutions

In \cite{CMS}, Chen-Ma-Salani established the strict convexity of spacetime level sets of solutions to heat equation in convex rings, using the constant rank theorem and a deformation method. In this paper, we generalize the constant rank theorem in \cite{CMS} to fully nonlinear parabolic equations, that is, establish the corresponding microscopic spacetime convexity principles of spacetime level sets. In fact, the results hold for fully nonlinear parabolic equations under a general structural condition, including the $p$-Laplacian parabolic equations ($p>1$) and some mean curvature type parabolic equations.