A NOTE ON THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS

The mean curvature flow is a fundamental geometric evolution equation that describes the motion of a hypersurface in a Riemannian manifold. It is a nonlinear parabolic partial differential equation that evolves the hypersurface according to the mean curvature at each point. In this note, we will discuss the mean curvature flow in Riemannian manifolds, its properties, and some important results related to its behavior.

Let $M$ be an $n$-dimensional Riemannian manifold with metric $g$. Given a smooth hypersurface $Sigma$ in $M$, the mean curvature vector $H$ at each point $p in Sigma$ is defined as the trace of the shape operator of $Sigma$ at $p$. The mean curvature flow is then given by the following partial differential equation:

where $X$ is a smooth map from $Sigma times [0, T)$ to $M$, and $T$ is the maximal existence time. The mean curvature flow can be seen as a heat flow on the hypersurface $Sigma$, where the curvature acts as a diffusion coefficient.

• Short-time existence: For a given initial hypersurface $Sigma_0$, there exists a short time $epsilon > 0$ such that the mean curvature flow exists for $t in [0, epsilon)$.

• Regularization effect: The mean curvature flow tends to smooth out the hypersurface, eliminating singularities and improving its regularity.

• Volume-preserving property: If the initial hypersurface has constant volume, then the volume of the hypersurface remains constant under the mean curvature flow.

• Convergence to minimal surfaces: Under certain conditions, the mean curvature flow converges to a minimal hypersurface, which minimizes the area among all hypersurfaces homologous to the initial hypersurface.

• Huisken's monotonicity formula: Huisken showed that the quantity $int_{Sigma} H^2$ is non-increasing along the mean curvature flow. This formula is key in studying the behavior of the flow.

• Hamilton's compactness theorem: Hamilton proved that if the mean curvature flow remains uniformly bounded in a sequence of hypersurfaces, then the sequence converges smoothly on compact subsets to a limit hypersurface.

• Mean convexity and singularities: In the case of mean convex initial hypersurfaces, the mean curvature flow develops singularities in finite time. The behavior of these singularities is a topic of active research.

The mean curvature flow is a rich area of study in differential geometry and geometric analysis, with connections to minimal surfaces, geometric measure theory, and geometric flows. It has applications in various fields, including mathematical physics, image processing, and materials science. Understanding the mean curvature flow in Riemannian manifolds is essential for exploring its theoretical properties and practical applications.