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2. Problems 10-18 suggested on day 1

This section includes problems 1-9 suggested on day 1.


    1. Add remark

      Problem 2.1.

      Is some positivity condition of Calabi curvature operator preserved by Kähler Ricci flow? Is it preserved by the Ricci flow ODE (need to check)?


        • Add remark

          Problem 2.2.

          Can we use Bochner formula for quaternionic lines to get results?


            • Add remark

              Problem 2.3.

              Is there a Tachibana Theorem that uses Calabi curvature operator?


                • Add remark

                  Problem 2.4.

                  How to directly link curvature operator of the first kind and curvature operator of the second kind (already done in the quaternionic Kähler case and the 3D case)? We probably need more than just writing down the eigenvalues and eigenspaces of the operators, as that does not encode the first Bianchi identity.


                    • Add remark

                      Problem 2.5.

                      Can we establish Tachibana theorem for $\nabla R$?


                        • Add remark

                          Problem 2.6.

                          Find a different proof of the following Bourguignon theorem: if $M^4$ has harmonic curvature tensor and nonzero signature $\sigma(M)\not=0$, then it is Einstein.


                            • Add remark

                              Problem 2.7.

                              Does $sec> \frac{n-1}{n+2}\frac{scal}{n(n-1)}$ implies that the manifold is homeomoephic to $\mathbb{S}^n/\Gamma$? This is already proven in dimension 4. Also, need to check if this statement generalizes $1/4$-pinching. It is also an optimal statement since equality holds for $ \mathbb{C}P^n$.


                                • Add remark

                                  Problem 2.8.

                                  Consider a closed spin manifold $(M,g)$. Take an arbitrary tensor $T$ and a spinor $s$. We then wish to apply the Bochner formula to $\omega=T\otimes s$. When is $Ric_L(\omega)>0$. Do we want an extra Kähler assumption here? Maybe we want to work with special holonomy?


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                                      Problem 2.9.

                                      Here we consider integral curvature operator bounds and the integral version of $k$-positivity curvature operator. Define a function $f:(M,g)\rightarrow\mathbb{R}$ by $f=\lambda_1+\cdot\cdot\cdot+\lambda_k$, where $\lambda_i$ is the $i$-th smallest eigenvalue of $\mathfrak{R}$. Can we find a bound on $\int_{(M,g)}f^p$ such that this bound gives us topological restrictions on $M$ or Bochner theorem results? Note that there is already a bound on the Betti numbers.

                                          Cite this as: AimPL: The Bochner technique, available at http://aimpl.org/bochner.