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1. Ricci Flow

1.     Consider $(S^3,g)$ with scalar curvature $R\geq 6$. The Ricci Flow exists on the maximal interval $(0,T)$. The scalar curvature evolves by $$\partial_t R = \Delta R + 2 |Rc|^2.$$ One can estimate that $$\partial_t R = \Delta R + 2 |Rc|^2 = \Delta R + \frac 2 3 R^2 + |\mathring{Rc}|^2 \geq \Delta R + \frac 2 3 R^2 .$$ It follows from the maximum principle that $$R\geq \frac{6}{1-4t} \mbox{ and } T \leq \frac 1 4.$$

Problem 1.05.

[D. Maximo, R. Bamler] If $T$ is close to $\frac 1 4$, is $g(0)$ close to the round metric in the Gromov-Hausdorff sense and in the $C^0$ sense? One can also ask the same question for other flows such as mean curvature flow. What would be a suitable topology?
1. Remark. ndavj
• Problem 1.1.

[B. Wilking] Let $K$ be a simplicial complex with metric $g$ such that each simplex has a metric with positive (or flat) curvature and $K$ has positive (or nonnegative) curvature in the Alexander sense. Does a Ricci flow $(K, g(t))$ exist such that for each $t>0$, $g(t)$ is a smooth orbifold metric and $$\lim_{t\to 0} g(t) =g$$ in the Gromov-Hausdorff topology. Can one get rigidity in the nonnegative case?
• Problem 1.15.

[B.Kotschwar] Does there exist a complete Ricci Flow that starts with uniformly bounded curvature that instantly has unbounded curvature?
• Problem 1.2.

[J. Streets] Given a hypersurface $M\subset \mathbb{R}^{n+1}$, define $$F(M,x_0,t_0)=(4\pi t_0)^{-\frac n 2} \int_M e^{-\frac{|x-x_0|^2}{4t_0}} dH^{n},$$ $$\lambda(M) = \sup_{(x_0,t_0)} F(M,x_0,t_0).$$ Question: Is there a useful notion of $\mu$-stability for Ricci Flow self-shrinkers analogous to Colding-Minicozzi for the $\lambda$ entropy in mean curvature flow? What are the stable solitons?
• Problem 1.25.

[R. Bamler] Understand singularities in Ricci Flow in dimension $n\geq 4$. Examples are $S^3\times \mathbb{R}$, $S^2\times \mathbb{R}^2$, $4d$ Bryant soliton, $3d \mbox{Bryant soliton} \times \mathbb{R}$ and FIK solitons.
• Problem 1.3.

[R. Bamler] In dimension $4$, can there be an ALE space as singularity for Ricci Flow?
• Problem 1.35.

[R. Bamler] Does $\max_M R(\cdot,t)$ blow up as $t\to T$ the singular time?
• Problem 1.4.

[R. Bamler] Let $(M^n,g_t)_{t\in[0,T)}$ be a Ricci flow with $T<\infty$. Assume $R <1$ on $M\times [0,T)$. Characterize $g_t$ as $t\to T$.
• Problem 1.45.

[R. Bamler] Let $(M^n,g_t)_{t\in[0,T)}$ be a Ricci flow with $T<\infty$. Assume $R< \frac{C}{T-t}$. Characterize $(T-t)^{-1} g_t$ as $t\to T$. This is related to the Hamilton-Tian conjecture. Note that the Kahler Ricci flow on Fano manifolds satisfies $R< \frac{C}{T-t}$.
• Problem 1.5.

[R. Bamler] Classify singularities modulo singularities that in bounded scalar curvature setting.
• Problem 1.55.

[R. Bamler] R. Bamler and Q. Zhang proved a backwards pseudolocality theorem: Assume $R<1$. For $r \in (0,1)$, if $|Rm| < r^{-2}$ on $B(x,t,r)$, then $$|Rm| < (\epsilon r)^{-2}$$ on $P(x,t,\epsilon r,-(\epsilon r)^2 ) =B(x,t,\epsilon r) \times [t-(\epsilon r)^2, t)$. Can we remove $R<1$ assumption in the backwards pseudolocality theorem?
• Problem 1.6.

[R. Bamler] Can we remove $R<1$ assumption in R. Bamler and Q. Zhang’s heat kernel bound and distance bound? For more information, see their papers on arxiv: HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATURE, and HEAT KERNEL AND CURVATURE BOUNDS IN RICCI FLOWS WITH BOUNDED SCALAR CURVATURE-PART II.

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