Singularity Analysis in Mean Curvature Flow

In this thesis, we investigate the formation of singularities in mean curvature flow. Specifically, we study ancient asymptotically cylindrical flows, i.e. ancient solutions whose tangent flow at $-\infty$ is a round shrinking cylinder $\mathbb{R}^{k}\times S^{n-k}(\sqrt{2(n-k)|t|})$, where $1\leq k\leq n-1$. While in the neck case, i.e. for $k=1$, a complete classification has been obtained in several breakthroughs, a classification for the case $2\leq k\leq n-1$ until recently seemed out of reach. To analyze ancient asymptotically cylindrical flows for $2\leq k\leq n-1$, we consider the cylindrical profile function $u$ that measures the deviation of the renormalized flow from the round cylinder. We prove that for $\tau\to -\infty$ we have the asymptotics $u(y,\omega,\tau)= (y^\top Qy -2\textrm{tr}(Q))/|\tau| + o(|\tau|^{-1})$, where $Q$ is a constant symmetric $k\times k$-matrix whose eigenvalues are quantized to be either 0 or $-\sqrt{(n-k)/8}$. We then focus on the extremal rank cases. Under the natural noncollapsing condition, we obtain a classification of all solutions with $\textrm{rk}(Q)=0$, and establish $\textrm{SO}(n-k+1)$-symmetry and unique asymptotics in the case $\textrm{rk}(Q)=k$, also known as the $k$-oval case. Next, we confirm a conjecture by Angenent-Daskalopoulos-Sesum about uniqueness of $\textrm{O}(k) \times \textrm{O}(n-k+1)$-symmetric ancient ovals and more generally classify all $\textrm{O}(k) \times \textrm{O}(n-k+1)$-symmetric ancient noncollapsed solutions. On the other hand, for every $2\leq k\leq n-1$ we construct a $(k-1)$-parameter family of ancient ovals that are only $\mathbb{Z}^{k}_{2}\times \mathrm{O}(n-k+1)$-symmetric, giving counterexamples to another conjecture of Daskalopoulos. We then investigate ancient ovals without any symmetry assumption. Specifically, we prove that any $2$-oval in $\mathbb{R}^4$, up to scaling and rigid motion, either is the unique $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by White and Haslhofer-Hershkovits, or belongs to our new one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar. Finally, as an application of the theory of ancient solutions, we prove that for the mean curvature flow of closed embedded hypersurfaces the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In particular, our analysis yields sharp curvature bounds improving prior results by Head and Cheeger-Haslhofer-Naber.

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