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3. Problems 19-23 suggested on day 1

This section includes problems 19-23 suggested on day 1.


    1. Add remark

      Problem 3.1.

      Can we do Bochner technique using 2 different Riemannian metircs? Llarull did the version for scalar curvature. Namely, Llarull showed that metrics $g$ on $\mathbb{S}^n$ that has $scal_g\geq scal_{round~metric}$ and $Area_g\geq Area_{round~metric}$ must be isometric to the round metric itself. Can we prove similar results for curvatures other than the scalar curvature?


        • Add remark

          Problem 3.2.

          Can we explore more about stability operator of high codimension minimal surfaces? Little is known about stability of high codimension minimal surfaces. Work has been done in low dimensions by Schoen-Yau.


            • Add remark

              Problem 3.3.

              For the Einstein case, we have both: $$\textrm{the Weitzenböck formula}: \Delta R=\frac{1}{2}Ric_L(R)$$ $$\textrm{the Ricci flow formula}: \Delta R+2(R^2+R^{\#})=2\lambda R.$$ Similarly in the gradient Ricci soliton case, we have both $$\textrm{the Weitzenböck formula}: \Delta R-\nabla_{\nabla f}R=\frac{1}{2}Ric_L(R)+\frac{1}{2}(\mathrm{Hess}f)R$$ $$\textrm{the Ricci flow formula}: \Delta R-\nabla_{\nabla f}R+2(R^2+R^{\#})=2\lambda R.$$ Is there some connection between the two formulas? Can we give an elliptic proof of the classification of gradient Ricci soliton with nonnegative curvature operator? We wish to understand the relationship between the two equations better.

              Furthermore, what is the corresponding evolution equation for $\overset{\circ}{R}$ under the Ricci flow? What does it imply in the Einstein case? Is there a version of Bochner technique formula for $g(\Delta R,R)$ that involves $\overset{\circ}{R}$?


                • Add remark

                  Problem 3.4.

                  Via the determinant function, we know that there is a natural embedding $\Lambda^nV\hookrightarrow \textrm{Sym}^p(End(V))$. Can we find a similar embedding for general $p$-forms with p<n, where we can do Bochner technique on the ambient vector space? Can we say something about cohomology/$p$-forms using sum of eigenvalues of the curvature operator? Or just using some suitable curvature conditions?


                    • Add remark

                      Problem 3.5.

                      We know that when $p=2$, PIC implies that $g(Ric_L(\omega),\omega)>0$ for all nonzero $\omega\in\Omega^2$. When $p>2$, is there an analogous condition we can put on the left hand side?

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