We show that the mean curvature flow of generic closed surfaces in\(\mathbb {R}^{3}\)
avoids asymptotically conical and non-spherical compact singularities. We also show that …

We prove that sufficiently low-entropy closed hypersurfaces can be perturbed so that their
mean curvature flow encounters only spherical and cylindrical singularities. Our theorem …

In this paper, we study the properties of nondegenerate cylindrical singularities of mean
curvature flow. We prove they are isolated in spacetime and provide a complete description …

We investigate slowly converging solutions for non-linear evolution equations of elliptic or
parabolic type. These equations arise from the study of isolated singularities in geometric …

R. Thom's gradient conjecture states that if a gradient flow of an analytic function converges
to a limit, it does so along a unique limiting direction. In this paper, we extend and settle this …

We prove a Liouville theorem for ancient solutions to the supercritical Fujita
equation\[\partial_tu-\Delta u=| u|^{p-1} u,\quad-\infty< t< 0,\quad p>\frac {n+ 2}{n-2},\] which …

We study the multiplicity of the singularity of mean curvature flow with bounded mean
curvature and Morse index. For $3\leq n\leq 6$, we show that either the mean curvature or …

Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares
many features with Hamilton's Ricci flow from intrinsic geometry. In this lecture series, I will …

In this thesis, we investigate the formation of singularities in mean curvature flow.
Specifically, we study ancient asymptotically cylindrical flows, ie ancient solutions whose …