1. Open Problems

Problem 1.05.
[Cheikh Ndiaye] Suppose we are given a closed 4manifold $(M^4, [g])$ which is not conformally equivalent to the standard 4sphere, 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 Qcurvature 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 GMT (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 codimension. Dimension of $X_+$ is even, dimension of $Y$ is odd. 
Problem 1.2.
[Jeffrey Case] Consider a closed 6manifold $M^6$. The space $\mathcal{S}$ of global conformal invariants is 4dimensional: $\mathcal{S}= \text{span}\{I_1,\cdots, I_4\}$.
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 $[g]_{k1} = \{g_w = e^{2w}g_{eucl}: Vol(g_w) = 1, A_w\in \Gamma_{k1}^+\}$.
Is \[ \inf_{g\in[g]_{k1}}\int \sigma_k(g_w^{1}A_w)dv_{g_w} \] attained? In particular, can this be proved using concentrationcompactness 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 plaplace 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 \[ Y_1(X,\partial X, [g]) = \inf_{\tilde g \in [g]} \frac{\int_X R_{\tilde g} \;dv_{\tilde g} + c_n\int_{\partial X}H_{\tilde g} \; dS_{\tilde g}}{(Vol(X,\tilde g))^{\frac{n1}{n+1}}} \] \[ Y_2(X,\partial X, [g]) = \inf_{\tilde g \in [g]} \frac{ \int_X R_{\tilde g} \;dv_{\tilde g} + c_n\int_{\partial X}H_{\tilde g} \; dS_{\tilde g}}{Vol(\partial X, \tilde g)^{\frac{n1}{n}}} \]
Consider $(X^{n+1},g_+)$ CCE, $[g_o] = $ conformal infinity, and assume $Y(\partial X,[g_o])>0$.
(ChenLaiWang) $Y_2\ge c(n)Y(\partial X,[g_o])$
(GurskyHan) $Y_1\ge c_1(n) (\mbox{isoperimetric ratio)} \cdot Y(\partial X, [g_o])$
(ChangGe) $Y_1\ge d_nY(\partial X,[g_o])$ (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 unknotted 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 4manifold $(M^4,[g])$ with $\partial M^4\not = \emptyset$ and a 2dimensional 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.