References for mean curvature flow in Riemannian manifolds

Question

I am interested in mean curvature flows (MCFs) in Riemannian manifolds. But most textbooks about MCF seem to treat mainly MCFs of hypersurfaces in $\mathbb{R}^{n+1}$. Should I study them before learning about MCFs in Riemannian manifolds? or Are there any textbooks about MCF in Riemannian manifolds for beginners?

Answer 1

You are right, there are not many books on MCF in a Riemannian manifolds. There are basically two reasons:

(1) Not much is known in the general case. One early result was given by G. Huisken, which says that if the curvature of the ambient space is pinched and the initial embedding is convex, then the MCF shrinks to a round point (This is a generalization of one of his first paper in MCF in 1984). This result can be found in Xiping Zhu's book.

(2) The singularity models all lie in $\mathbb R^n$. If $M$ is embedded in $N^n$ and $M$ is compact, then general theory of parabolic PDE tells us that MCF can be defined for some short time $t$. In general singularity will occur at some point $p$ in the ambient space. To study the singularity, we normally blowup at point and in the limit that will corresponds to a MCF in $\mathbb R^n$ (a self shrinker, an translator etc., depending on the blowup procedure and the type of singularity)