7 problems

Let $M$ be a compact, locally irreducible Riemannian manifold that is rigidly scalar-flat, meaning that it admits a scalar-flat metric but no metric of positive scalar curvature. I…

Let $\varphi$ be a closed $G_2$-structure, meaning a positive closed three-form, on a closed $7$-manifold. Consider the Laplacian flow … A closed $G_2$-structure has sufficiently s…

Let $M_\zeta$ be the ALE resolution constructed from the quotient $V/\Gamma$, and let $\omega_\zeta$ denote its Kähler form. The Calabi–Yau ALE metric conjecture. The ALE space…

Fix $\lambda<0$ and consider the 4-dimensional space of local $\textup{SU}(3)$-invariant shrinkers, together with the complete AC shrinkers among them on $\Lambda^2_-\mathbb{CP}^2$…

Let $\Lambda^2_-\mathbb{S}^4$ be the anti-self-dual bundle of the 4-sphere, and let the explicit AC shrinker be the complete $\textup{Sp}(2)$-invariant shrinker described in Exampl…

Let $\Lambda^2_-\mathbb{CP}^2$ denote the anti-self-dual bundle of the complex projective plane. A complete expander has quadratic-exponential volume growth when its volume grows a…

Consider complete AC shrinkers with $\textup{SU}(3)$ symmetry on the relevant cohomogeneity-one $\textup{SU}(3)$-invariant geometry. SU(3)-invariant shrinker-finiteness conjecture.…