Convexity of Surfaces Moving by Mean Curvature Flow

This thesis investigates when convexity (non-negativity of the principal curvatures, resp. of the second fundamental form $h_{ij}$) is preserved under Mean Curvature Flow in general Riemannian manifolds.

Key Points

Setup & Tools: Development of the necessary differential geometry (embeddings, second fundamental form $h_{ij}$, Gauss/Codazzi equations, Simons identity) and derivation of central evolution equations under Mean Curvature Flow — among others for $h_{ij}$, the mean curvature $H$, and the norm of the Weingarten map $|A|^2$.

Main Result (after Huisken 1986): In curved ambient spaces, “naive” convexity ($h_{ij} \geq 0$) is not automatically invariant. Instead, a modified convexity condition is formulated that takes into account bounds on the ambient curvature $K_1$ and its derivative $L$; this modified convexity is preserved under the flow (maximum principle for tensors as the key technique).

Additional Chapter: Presentation of Huisken’s pinching result: the principal curvatures $\kappa_i$ approach each other in the course of the flow, i.e. $|A|^2 - \frac{H^2}{n-1} \to 0$ (“pinching”).

Rigidity / Examples: Two constructive examples show that the modification is strictly necessary:

1. In hyperbolic space, an initially convex surface ($h_{ij} \geq 0$) can immediately lose convexity — influence of the ambient curvature ($K_1 > 0$, $L = 0$).
2. In Berger spheres, convexity can be lost locally — influence of the curvature gradient $\nabla \bar{R}$ ($K_1 > 0$, $L > 0$).