Planarity and the mean curvature flow of pinched submanifolds in higher codimension

In this thesis, we explore the role of planarity in mean curvature flow in higher codimension and investigate its implications for singularity formation in a certain class of flows. In Chapter 1, we show that the blow-ups of compact 𝑛-dimensional solutions to mean curvature flow in ℝⁿ initially satisfying the pinching condition |𝐴|² < c |𝐻|² for a suitable constant c = c(𝑛) must be codimension one. We do this by establishing a new a priori estimate via a maximum principle argument. In Chapter 2, we consider ancient solutions to the mean curvature flow in ℝⁿ⁺¹ (𝑛 ≥ 3) that are weakly convex, uniformly two-convex, and satisfy derivative estimates |∇𝐴| ≤ 𝛾1 |𝐻|², |∇² 𝐴| \leq 𝛾2 |𝐻|³. We show that such solutions are noncollapsed. The proof is an adaptation of the foundational work of Huisken and Sinestrari on the flow of two-convex hypersurfaces. As an application, in arbitrary codimension, we classify the singularity models of compact 𝑛-dimensional (𝑛 ≥ 5) solutions to the mean curvature flow in ℝⁿ that satisfy the pinching condition |𝐴|² < c |𝐻|² for c = min {1/𝑛-2, 3(𝑛+1)/2𝑛(𝑛+2)}. Using recent work of Brendle and Choi, together with the estimate of Chapter 1, we conclude that any blow-up model at the first singular time must be a codimension one shrinking sphere, shrinking cylinder, or translating bowl soliton. Finally, in Chapters 3 and 4, we prove a canonical neighborhood theorem for the mean curvature flow of compact 𝑛-dimensional submanifolds in ℝⁿ (𝑛 ≥ 5) satisfying a pinching condition |𝐴|² < c |𝐻|² for $c = min {1/𝑛-2, 3(𝑛+1)/2𝑛(𝑛+2)}. We first discuss, in some detail, a well-known compactness theorem of the mean curvature flow. Then, adapting an argument of Perelman and using the conclusions of Chapter 2, we characterize regions of high curvature in the pinched solutions of the mean curvature flow under consideration.