• Corpus ID: 203952196

Singularity models of pinched solutions of mean curvature flow in higher codimension

@article{Naff2019SingularityMO,
  title={Singularity models of pinched solutions of mean curvature flow in higher codimension},
  author={Keaton Naff},
  journal={arXiv: Differential Geometry},
  year={2019}
}

• Keaton Naff
• Published 9 October 2019
• Mathematics
• arXiv: Differential Geometry

We consider noncompact ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) that are strictly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq \gamma_1 |H|^2, |\nabla^2 A| \leq \gamma_2 |H|^3$. We show that such an ancient solution must the translating bowl soliton. As an application, in arbitrary codimension, we consider compact $n$-dimensional ($n \geq 5$) solutions to the mean curvature flow in $\mathbb{R}^N$ that satisfy the pinching condition $|H| > 0$ and $|A|^2 < c(n) |H|^2$, $c(n) = \min\{\frac{1}{n-2}, \frac{3(n+1)}{2n(n+2)}\}$. 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.