9 problems

Let $S^{2}$ be the two-sphere, let $\mu$ be the measure used in the definitions of the functionals $A_{n}$ and $B_{n}$, and let $A_{n}(\varphi)$ and $B_{n}(\varphi)$ be the corresp…

Let $C$ be a smooth closed curve with canonical constant-curvature metric $g_{0}^{C}$, and let $l$ be a holomorphic line bundle with induced canonical metric $g_{0}^{l}$. For a Her…

Let $C$ be a compact complex curve with Hermitian metric $g^{C}$, and let $l$ be a line bundle over $C$ with associated Hermitian metric $g^{l}$. Write $A(g^{C},g^{l})$ for the log…

Let $(M,\omega_{KE})$ be a Kähler–Einstein manifold with positive first Chern class and $\operatorname{Ric}(\omega_{KE})=\omega_{KE}$, and let $P(M,\omega_{KE})$ be the space of no…

Existence conjecture. For every $\Lambda\in(4\pi,+\infty)$, $F(u)$ has at least one critical point subject to this constraint. This would differ substantially from the non-existenc…

Let $\Omega$ be a simply connected domain, and consider the Moser-Trudinger functional … under the Dirichlet-energy constraint $\int_{\Omega}|\nabla u|^2dx=\gamma$. For sufficientl…

McLeod–Peletier conjecture. There should exist a function $g$ satisfying these conditions such that $C_{g,4\pi}(\Omega)$ does not admit any extremal function. The conjecture is pos…

Let $\Omega\subset\mathbb R^2$ be a regular, bounded and convex domain, and define … for $u\in H^1_0(\Omega)$, where $d(x,\partial\Omega)$ denotes the distance from $x$ to the boun…

Tian's exponent-one conjecture. The exponent $\gamma$ in Tian's Moser–Trudinger inequality should be admissible with $\gamma=1$.