40 problems

Let $(M^{n},g_{0})$ be a compact Riemannian manifold of dimension $n\ge 3$. Consider the Yamabe equation … where $R_{g_0}$ is the scalar curvature. Yamabe compactness conjecture. T…

Schoen's conjecture. At a blow-up point, the Weyl tensor should vanish up to $[\frac{n-6}{2}]$-th order derivatives.

In conformal geometry, let $(M,g)$ be a smooth, compact, aspherical Riemannian manifold, and consider the set of Yamabe metrics on $M$. Compactness Conjecture. This set is compact.…

Let $(M,g_0)$ be a smooth compact $n$-dimensional Riemannian manifold with boundary $\partial M$, where $n\geq 3$, and let $Y(M,\partial M,[g_0])$ denote its generalized Yamabe con…

Let $M$ be a compact quaternionic contact manifold that is locally non-flat in the conformal quaternionic contact sense. Let $\lambda(M)$ denote its qc Yamabe constant, and let…

Let $S^{4n+3}$ be the standard unit $(4n+3)$-dimensional quaternionic sphere with its standard quaternionic contact structure and standard 3-Sasaki structure. Its qc-scalar curvatu…

Fully nonlinear Yamabe conjecture. There exists a positive smooth function $u\in C^\infty(M^n)$ such that

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 $S^{2n+1}$ be the CR sphere with standard contact form … The CR automorphisms of $S^{2n+1}$ are those induced by biholomorphisms of the unit ball in $\mathbb{C}^{n+1}$. Jerison…

Andrade–Piccione–Wei conjecture. Under these hypotheses, there is a positive minimizer for $Y_{Q_6}(M^n,[g])$. This concerns existence of a minimizing metric for the sixth-order Ya…

Let $\psi$ be a ground state solution of the limit equation … where $\lambda$ is the parameter in the equation and the Pauli matrices $\gamma_1,\gamma_2,\gamma_3$ are as defined in…

Let $(M,g)$ be a compact Riemannian manifold with boundary, let $Y(M,\partial M,[g])$ denote the Yamabe invariant, and consider Equation … Y(M,partial M,[g])0, … admits a positive…

Scalar-curvature sign-change conjecture. When $[?]=0$, a prescribed function $S$ can occur only if it changes sign and

Boundary Yamabe critical-dimension conjecture. The critical dimension for compactness of the full set of solutions to the boundary Yamabe problem is $n=24$; in particular, compactn…

Schoen–Yau conjecture. Any distributional solution to the associated partial differential equation would provide a complete metric for the Yamabe problem.

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…

Let ${\mathbb S}^n_a=(\mathbb{S}^n,h_a)$ be the $n$-dimensional sphere endowed with a standard edge-cone spherical metric of cone angle $\alpha=2\pi a$ along a great circle…

Let $(M^m,g)$ be a Riemannian manifold and consider the product $(M\times\mathbb{R}^n,g+dx^2)$. A positive finite-energy solution of the Yamabe equation on this product is understo…

Kazdan–Warner's conjecture. When $Y(M)=0$ and $f\not\equiv0$, the equation possesses a positive solution if and only if

Let $M$ be a compact Riemannian manifold with boundary $\partial M$, let $q_0\in\partial M$ be a blow-up point, let $\gamma$ be the linear perturbation term, and let the trace-free…