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2. Mean Curvature Flow

    1. Problem 2.1.

      [B. Kleiner] Let $C\subset \mathbb{R}^2$ be the topologist’s sine curve. More precisely, $$C=\{(x,\sin \frac 1 x )| x\in (0,\frac{1}{2 \pi})\} \cup \Gamma,$$ where $\Gamma$ is a smooth curve connecting the origin and the point $(\frac{1}{2 \pi}, 0)$ that doesn’t intersect $C\setminus \Gamma$. What happens to this when you apply level set flow? Does it become instantly smooth? More generally, what happens to a compact set $\Omega \subset \mathbb{R}^2$ under level set flow?

      It is known that Jordan curves of zero Lebesgue measure become instantly smooth. It is also known that any compact connected set $\Omega$ with Lebesgue measure $H^2(\Omega)=0$ is nonfattening, provided it separates $\R^2$ into precisely two connected components. The level set flow of an arbitrary compact locally connected set is pretty well understood. See Joseph Lauer’s paper on the arxiv for these facts and other background.
        • Problem 2.2.

          [X.H. Nguyen] For $d=1$, if we have a bounded $C^{2,\alpha}$ graph, let $c(t)$ be the curve shortening flow and let $u(t)$ be the solutions to be heat equation. Nara and Taniguchi proved that $$|c(t)-u(t)| \to 0 \mbox{ as } t \to \infty.$$ Is there a higher dimension analog? The rotationally symmetric case is known.
            • Problem 2.3.

              [B. Kleiner] Pick an integer $N$, and onsider a smooth embedded curve $\gamma\subset\R^2$ constructed as follows. Let $\alpha_\pm:[0,2\pi N]\rightarrow\R^2$ be a pair of disjoint embeddings that admit polar coordinate parametrizations $t\mapsto (r_\pm(t),\theta_\pm(t)) $ where $r_\pm$ is increasing and has $C^2$ norm $<\frac{1}{N}$, and $\theta_\pm(t)=t$. Thus $\alpha_\pm$ wraps around the unit circle $N$ times, spiralling slowly outward. Now concatenate $\alpha_+,\alpha_-$ with a pair of curves $\beta_+,\beta_-$ of curvature and diameter $<10$ to form the curve $\gamma$.

              What happens to $\gamma$ under curve shortening flow? It is known that it will shrink to a point, and become asymptotically round — the problem is to provide a quantitative narrative for how this occurs. This should include estimates on how the length and total curvauture behave as a function of time, and a geometric description of the structure of the curve.
                • Problem 2.4.

                  [B. Kleiner] Recall that the mean curvature flow of a “thin” torus of revolution in $\R^3$ goes singular everywhere simultaneously as it converges to a round circle. Under what conditions on a smooth embedded closed curve $\gamma\subset\R^3$ does there exist a toroidal mean convex mean curvature flow $M_t$ that Hausdorff converges to $\gamma$ at the blow-up time?

                      Cite this as: AimPL: Geometric flows and Riemannian geometry, available at http://aimpl.org/flowriemannian.