16 problems

For a hypersurface with prescribed $k$-th elementary symmetric curvature, let $k$ denote the order of the higher-order mean curvature, with $k=1$ corresponding to mean curvature an…

Let $(M,g)$ satisfy equation … left|Rmright|{L^2(B(p0,2r))}leqepsilon0. … CS(p0,r/2)leq C V0^{-1/4}, … , $\epsilon_0$ can be chosen independently of $V_0$, and the volume lower-bou…

Nitsche–Hopf conjecture. For every minimal graph over a disk,

Let $0 \leq l < k \leq n$. Let $\mathbb{H}^{n+1}$ be hyperbolic space, and let $\Gamma=\{\Gamma_1,\ldots,\Gamma_m\}\subseteq\partial_{\infty}\mathbb{H}^{n+1}$ be a disjoint collect…

Let $n$ and $k$ satisfy $2k>n$, and let $\kappa=(\kappa_1,\dots,\kappa_n)\in\Gamma_k$, where $\Gamma_k$ is the Gårding cone for the $k$-th elementary symmetric polynomial. Assume t…

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

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

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

Epsilon-regularity conjecture. In these extremal Kähler and constant-scalar-curvature half-conformally flat cases, Corollary Harnack and part (iii) of the epsilon-regularity theore…

Let $(\Sigma_0,g,k)$ be an initial data set for the Einstein-vacuum equations, where $g$ is the Riemannian metric on the initial hypersurface and $k$ is its second fundamental form…

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…