9 problems

Ren–Wang's concavity conjecture. For every $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb{R}^n$,

Uniform curvature bound conjecture. The curvature remains uniformly bounded up to the maximal time:

The quadratic-form conjecture. For every $n4$-dimensional vector $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb{R}^n4$, the quadratic form

Bounded $L^2$-curvature conjecture. The Einstein scalar field equations admit local Cauchy developments for initial data sets $(\Sigma,g,k,u_{0},u_{1})$ with locally finite $L^{2}$…

Let $\{M_t\subset\mathbb{R}^{n+1}\}_{t\in [0,T)}$ be a mean curvature flow of two-convex closed embedded hypersurfaces. Here $\lvert A\rvert$ denotes the norm of the second fundame…

Immersed curvature estimate conjecture. The uniform curvature estimate

All-dimensional $L^2$ curvature bound conjecture. There exists $C=C(n,{\rm v})>0$ such that

Curvature and rescaling conjecture. There exists $C>0$ such that, for all $t\in(0,1]$,

Let $M^4$ be a Riemannian manifold, let $B_{r_0}=B_{r_0}(x)\subset M^4$, and let $\Sigma^3\subset B_{r_0}$ be an immersed $2$-sided stable minimal hypersurface with…