5 problems

Let $M$ be a closed smooth manifold, and let $g$ be an $L^\infty$-metric on $M$, meaning that for some smooth reference Riemannian metric $g_0$ and constant $\Lambda>0$,…

Burgos-Kramer-Kühn's conjecture. This metric corresponds to a nef toric b-divisor, which in turn corresponds to the compact convex set

Lee's positive mass conjecture. The positive mass theorem holds for $g$; in particular, its ADM mass is nonnegative, with equality only in the appropriate flat rigidity case.

Let $f\colon X\to Y$ be a projective surjective morphism between complex manifolds. Let $(L,h)$ be a holomorphic line bundle over $X$ equipped with a singular Hermitian metric of s…

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…