| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

1. Problems 1-9 suggested during day 1

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


    1. Add remark

      Problem 1.1.

      Counterexample to some positive curvature condition involving $\overset{\circ}{R}$ being preserved under the Ricci flow.


        • Add remark

          Problem 1.2.

          Find a metric on $\mathbb{S}^n\times\mathbb{S}^n$ with $(n+1)$-positive curvature operator.


            • Add remark

              Problem 1.3.

              Calculate eigenvalues and eigenspaces of curvature operators (of the second kind) of more symmetric spaces/other homogeneous spaces.


                • Add remark

                  Problem 1.4.

                  Does $R(X,Y,\bar{X},\bar{Y})\geq \kappa$ and $diam(M,g)\leq D$ imply $Ric_L(\omega,\omega)>C(p,n,\kappa, D)|\omega|^2$ for all $\omega\in\Omega^p(M)$? Does $R(X,Y,\bar{X},\bar{Y})\geq \kappa$ and $diam(M,g)\leq D$ also imply $\hat{R}\geq C(p,n,\kappa, D)$?


                    • Add remark

                      Problem 1.5.

                      [Yau 1980] Consider two Kähler manifolds $M^n_1$ and $M^n_2$ with $sec_{M_i}\leq -\kappa_i<0$, and assume that we have a harmonic map $F:M_1\rightarrow M_2$ that has full rank at some point $p$, can we prove that $F$ is holomorphic or antiholomorphic?


                        • Add remark

                          Problem 1.6.

                          We know that $sec\geq 0$ implies that $g(Ric_L(T),T)\geq 0$ for all $T\in Sym^k(M)$. Can we find some constant $k(p)$ dependent on $p$, such that $p$-positive curvature operator implies $g(Ric_L(T),T)\geq 0$ for all $T\in Sym^{k(p)}(M)$?


                            • Add remark

                              Problem 1.7.

                              Does $g(Ric_L(T),T)\geq 0$ for all $T\in Sym^k(M)$ implies $g(Ric_L(S),S)\geq 0$ for all $S\in Sym^{k-1}(M)$?


                                • Add remark

                                  Problem 1.8.

                                  Consider complete gradient shrinking Kähler Ricci solitons. Can we prove that the Lie algebra of holomorphic vector fields is isomorphic to a conjugate of complexification of Killing vector fields?


                                    • Add remark

                                      Problem 1.9.

                                      Can we characterize Fubini-Study metric $\mathbb{C}P^{\frac{n}{2}}$ among Kähler Einstein manifolds under conditions involving curvature operator of the second kind or Calabi curvature operator? (a) Can we find a condition that makes $\mathbb{C}P^{\frac{n}{2}}$ the only example? (b) Can we find a sharp condition?

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