First, we review the authors’ recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.
We formulate a class of functionals in space forms such that its critical points include the r-minimal hyper-surface and the minimal hyper-surface as special cases. We obtain the algebraic, differential and variational characteristics of the critical surfaces determined by the critical points. We prove the Simons' type nonexistence theorem which indicates that in the unit sphere, there exists no stable critical surfaces, and the Alexandrov's type existence theorem which indicates that in Euclidean space, the sphere is the only stable critical surfaces.
