8 problems

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 $(M,J)$ 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…

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

Let $M^n$ be a closed, connected $n$-manifold, let $S^1$ be the circle, and let $\sigma$ denote the Yamabe invariant. Rosenberg's $S^1$-stability conjecture. The Yamabe invariants…

Yamabe-invariant equality conjecture. If