Loading Web-Font TeX/Main/Regular
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

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.

      Add remark

      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


            • Add remark

              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?


                • Add remark

                  Problem 1.15.

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


                    • Add remark

                      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?


                        • Add remark

                          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.


                            • Add remark

                              Problem 1.3.

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


                                • Add remark

                                  Problem 1.35.

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


                                    • Add remark

                                      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.


                                        • Add remark

                                          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}.


                                            • Add remark

                                              Problem 1.5.

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


                                                • Add remark

                                                  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?


                                                    • Add remark

                                                      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.