16 problems

Schoen's Weyl-vanishing conjecture. If $\overline{y}$ is a blow-up point, then the Weyl tensor $W_{g_0}$ of $g_0$ vanishes at $\overline{y}$ to order $\left[\frac{n-6}{2}\right]$;…

Compactness and existence conjecture. If $A_g\in\Gamma_k^+$, then there exists a conformal deformation $\tilde{g}=e^{2u}g$ such that

Let $k,m\in\mathbb{N}$ and let $N(k,m)$ be the dimension threshold appearing in the higher-order GJMS geometric Aubin set conjecture for conformal manifolds virtually conformal to…

The higher-order GJMS geometric Aubin set conjecture. The set $\mathscr{A}_{n,2k}$ is a geometric Aubin set for $L_{2k}$.

Let $(M,J,\theta)$ be a compact strictly pseudoconvex CR manifold of real dimension $2n+1$, and let $G$ be a compact subgroup of $I(M,\theta)$, the group of pseudo-Hermitian transf…

Let $T$ be a solid torus, i.e., a handlebody with one handle. Define … where the supremum is taken over all smooth, bounded domains $\Omega\subset\mathbb{R}^3$ whose closure is dif…

Let $x_1,\dots,x_p$ with $p\geq 2$ be distinct vectors in $\mathbb R^m$. Define the symmetric matrix $A=(a_{ij})_{1\leq i,j\leq p}$ by … For $k=1,\dots,p$, let…

Let $(M^n,g)$ be a smooth compact Riemannian manifold with $Ric\geq 0$ and second fundamental form $\Pi\geq 1$ on its boundary $\Sigma$. Let $u\in C^{\infty}(M)$ be a positive solu…

Uniqueness conjecture. Under these hypotheses, the normalized solution of the $v_m$-Yamabe equation is unique. The conjecture is suggested by the variational and geodesic-convexity…

Yamabe's conjecture. There exists a metric $g$ on $X$ conformally equivalent to $g_0$ and having constant scalar curvature.

Let $M$ be a nonempty manifold of dimension $n\geq 3$, and let $g\in\operatorname{Metr}(M)$. For a metric $\overline{g}$ conformal to $g$, write…

CR $k$-Yamabe existence conjecture. If

CR sphere k-Yamabe conjecture. If the pseudohermitian structure $\widetilde{\theta}=e^{2u}\hat{\theta}\in[\hat{\theta}]_{+}$ has constant pseudohermitian $k$-curvature, then…

Let $g$) be a smooth Riemannian metric defined on the unit half $n$-ball $B^+_1$, where $4\leq n\leq24$. Suppose there is a sequence of positive solutions $\{u_i\}$ of … with…

Lichnerowicz's conjecture. There exists a metric $\tilde g$ conformal to $g$, with constant scalar curvature $R_{\tilde g}$, such that

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$. For nonzero $\psi\in H_1(M)$, define the Yamabe functional … and set … Here $R_g$ is the scalar curvatur…