Nonlinear diffusion with a bounded stationary level surface

We consider (non-degenerate) nonlinear fast diffusion of some substance in a container (not necessarily bounded) with bounded smooth boundary. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at constant unit density. We show that, if the container contains a proper time-invariant level surface of the substance's density (a surface on which the density is constant at each given time and that is the boundary of a smooth subdomain), then the boundary of the container must be a sphere. The key ideas are two: 1) a short-time asymptotic formula for the substance's density that generalizes to the nonlinear case in hand one due to S.R.S. Varadhan; 2) a new application of the method of moving planes created by A.D. Aleksandrov and perfected by J. Serrin. The asymptotic formula is obtained by a scheme that relies on the theory of viscosity solutions and implies that the time-invariant level surface must be parallel to (its point must be at equal distance from) the boundary of the container. This fact allows the use of a new, simple and very general application of the method of moving planes. The benefits of this general scheme are then extended to other settings that include the case of the heat flow in the sphere and the hyperbolic space.