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

1. Open Problems

    1. Problem 1.05.

      [Cheikh Ndiaye] Suppose we are given a closed 4-manifold $(M^4, [g])$ which is not conformally equivalent to the standard 4-sphere, and for which $P_g$ is nonnegative, has trivial kernel, and $k_g = 8\pi^2$. Study the flow \[ \frac{\partial g}{\partial t} = (-P_gQ_g)g. \]

      Does a solution exist globally in time? Does it converge to a constant Q-curvature metric?
        • Problem 1.1.

          [Robin Graham] Let $(X^{n+1}, g_+)$ be a Poincaré Einstein manifold and assume $Y^{k+1}\subseteq X^{n+1}$, $k$ odd, is minimal and regular at infinity. Does there exist a local conformal invariant $\mathcal{I}$ such that \[ A= c_{k,n}\chi(Y) + \int_Y\mathcal{I}\;dS? \]
            • Problem 1.15.

              [Andrew Waldron] (A) Generalize formula of G-M-T (see below) to higher dimensions. \[ \pi^2(4\chi(X_+) - \chi(\Sigma)) = \frac{3}{4}V_+ + \frac{1}{8}\int_{X_+}|W|^2dv+ \int_Y C\;dS \]

              (B) Generalize (A) to higher co-dimension. Dimension of $X_+$ is even, dimension of $Y$ is odd.
                • Problem 1.2.

                  [Jeffrey Case] Consider a closed 6-manifold $M^6$. The space $\mathcal{S}$ of global conformal invariants is 4-dimensional: $\mathcal{S}= \text{span}\{I_1,\cdots, I_4\}$.

                  Theorem (Case, 2022): $I_1([g])\ge 0$ if $[g]$ admits an einstein metric. (arXiv:2207.00645 [math.DG])

                  Definition \[ I_i(M^6) := \sup_{[g]} I_i([g]) \]

                  (A) Does there exists a closed manifold $M^6$ such that $I_1(M^6)<0$?

                  (B) Does there exist a closed manifold $M^6$ with $\pi_1(M^6) = \infty$ (infinite fundamental group) and $c_1,c_2,c_3\ge 0$ such that $\sum_ic_iI_i(M^6)<0$?
                    • Problem 1.25.

                      [Yi Wang] Assume $k<\frac{n}{2}$, $n\ge 3$, and let $[g]_{k-1} = \{g_w = e^{-2w}g_{eucl}: Vol(g_w) = 1, A_w\in \Gamma_{k-1}^+\}$.

                      Is \[ \inf_{g\in[g]_{k-1}}\int \sigma_k(g_w^{-1}A_w)dv_{g_w} \] attained? In particular, can this be proved using concentration-compactness arguments? Can minimizers be classified? (Are they bubbles?)
                        • Problem 1.3.

                          [Jie Qing] Are there any “curvature quantity" which transforms under conformal changes via the p-laplace operator? Note: $\int |\nabla u|^n\;dv_g$ is conformally invariant. In general, is there a conformal invariant operator whose leading term is the $p$-laplace operator?
                            • Problem 1.35.

                              [Alice Chang] $X^{n+1}$ compact manifold with $\partial X^{n+1}\not = \emptyset$. Define \[ Y_1(X,\partial X, [g]) = \inf_{\tilde g \in [g]} \frac{\int_X R_{\tilde g} \;dv_{\tilde g} + c_n\int_{\partial X}H_{\tilde g} \; dS_{\tilde g}}{(Vol(X,\tilde g))^{\frac{n-1}{n+1}}} \] \[ Y_2(X,\partial X, [g]) = \inf_{\tilde g \in [g]} \frac{ \int_X R_{\tilde g} \;dv_{\tilde g} + c_n\int_{\partial X}H_{\tilde g} \; dS_{\tilde g}}{Vol(\partial X, \tilde g)^{\frac{n-1}{n}}} \]

                              Consider $(X^{n+1},g_+)$ CCE, $[g_o] = $ conformal infinity, and assume $Y(\partial X,[g_o])>0$.

                              (Chen-Lai-Wang) $Y_2\ge c(n)Y(\partial X,[g_o])$

                              (Gursky-Han) $Y_1\ge c_1(n) (\mbox{isoperimetric ratio)} \cdot Y(\partial X, [g_o])$

                              (Chang-Ge) $Y_1\ge d_nY(\partial X,[g_o])$ (no dependence on isop. constant)

                              What is the best $d_n$? Is it attained? (If so, characterize)
                                • Problem 1.4.

                                  [Andrea Malchiodi] Relate the Mobius energy of an un-knotted loop $\gamma$ in $\mathbb{S}^3$ to the renormalized area of a minimal disk $D\subseteq \mathbb{H}^4$ with $\partial D = \gamma$.
                                    • Problem 1.45.

                                      [Alice Chang] Suppose $(\mathbb{R}^{n+1}_+,g_+)$ satisfies: i) $Ric(g_+) = -ng_+$, ii) $y^2g_+$ extends to $\mathbb{R}^n\subset \mathbb{R}^{n+1}_+$ such that (iii) $y^2g_+|_{R^n} = e^{2f}g_{eucl}$. Under what condition(s) does it follow that $g_+ =g_{hyp}$?
                                        • Problem 1.5.

                                          [Stephen McKeown] Suppose we are given a 4-manifold $(M^4,[g])$ with $\partial M^4\not = \emptyset$ and a 2-dimensional corner $\Sigma$. Does there exist a unique $\tilde g = e^{2w}g$ such that \[ \begin{cases} Q_{\tilde g} &=0 \; \text{ in }M^4,\\ T_{\tilde g}&= 0 \; \text{ on }\partial M^4,\\ U_{\tilde g}& = \lambda \; \text{ on }\Sigma? \end{cases} \]
                                            • Problem 1.55.

                                              [Alice Chang] Let $g_o$ be a conformally compact metric defined on $\mathbb{B}^4$. Assume $Y(\mathbb{S}^3,[\rho^2 g_o])>0$, where $\rho>0$ is a defining function. Consider the normalized Ricci flow \[ \begin{cases} \frac{\partial g}{\partial t} &= -2g - 6\text{Ric}_g\\ g(0,\cdot) &= g_0. \end{cases} \] Analyze singularity formation.

                                                  Cite this as: AimPL: Partial differential equations and conformal geometry, available at http://aimpl.org/pdeconformal.