A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a single scalar equation whose solutions provide solutions of the original geometric problem by providing a suitable graph height function.

Once the scalar equation is obtained, the author simply appeals to the "standard theory of scalar parabolic PDE"; however, the `standard theorem' required is certainly not standard, as it needs to provide solutions of a fully non-linear, second order, strictly (not uniformly) parabolic PDE over a closed Riemannian manifold (not simply $\mathbb{R}^n$). So my questions are as follows:

1) Where can I find the above `theorem' (if it exists)?

2) I have seen the above theorem but with the uniform parabolicity assumption. Does the case I need (strictly, but not necessarily uniformly, parabolic) follow easily? Seems reasonable since the spatial manifold is compact.

Thanks for any help, ML