12 problems

Rosenberg's $S^1$-stability conjecture.

Let $M$ be a closed $3$-manifold that admits a constant-curvature metric $h$. Here $Y(M)$ denotes the Yamabe invariant of $M$, and $E(h)$ denotes the normalized Einstein-Hilbert en…

Let $N$ be a closed manifold undergoing a surgery deformation through manifolds $N_{\tau}$, where $N_{\tau}$ is diffeomorphic to $N$ for $\tau<0$ and to the surgered manifold…

Let $(M,J)$ be a compact complex surface, where $M$ is its underlying 4-manifold and $Y(M)$ is its Yamabe invariant. Negative Yamabe invariant conjecture. If … then $M$ is of gener…

Let $M$ be the underlying 4-manifold of a compact complex manifold of Kodaira dimension $0$ or $1$, and let $b_1(M)$ denote its first Betti number. Yamabe invariant conjecture for…

Let $M$ be the underlying 4-manifold of a complex algebraic surface of Kodaira dimension $\geq 0$. A sequence of unit-volume constant-scalar-curvature metrics on $M$ has uniformly…

$S^2\times S^2$ equality conjecture. Under that assumption, one might further conjecture that

Akutagawa's radiality conjecture. This infimum is achieved by a function radial in the $\mathbb{R}^{m_2}$ variable.

Yamabe product conjecture. The Yamabe constant $Y(g(T))$ is minimized by $u_{n,T}$, and

Yamabe-invariant equality conjecture. If

Let $S^n$ be the round $n$-sphere and let $\Gamma$ be a finite group acting freely and isometrically on $S^n$, so that $S^n/\Gamma$ is a finite quotient of the round sphere. Let…

Let $\Sigma$ be a three-manifold, let $Y(\Sigma)$ denote its Yamabe invariant, and let $k$ be the mean-curvature parameter of a CMC flow. The maximal range of $k$ is a connected op…