7 problems

Let $(M^n,g)$ be a complete non-compact asymptotically Euclidean Riemannian manifold, with ADM mass $m_\text{ADM}(g)$ and scalar curvature $\operatorname{Scal}^g$. Positive mass co…

Geometric inequality conjecture. One has

Asymptotically Euclidean immersion conjecture. There exists an asymptotically Euclidean isometric immersion

Let $(M^3,g)$ be an asymptotically Euclidean manifold of order $\tau\in(-1,-\frac{1}{2})$ with respect to a structure of infinity $\Phi_z$ and coordinates $\{z^i\}_{i=1}^3$. Assume…

Let $(M^3,g)$ be an asymptotically Euclidean manifold of order $\tau\in(-1,-\frac{1}{2})$ with respect to a structure of infinity $\Phi_z$ and coordinates $\{z^i\}_{i=1}^3$. Assume…

Let $(u_0,u_1)$ be nontrivial, nonnegative, compactly supported smooth initial data for the semilinear wave equation on an asymptotically Euclidean manifold, with small-data size ……

Positive mass conjecture. Then $m(g)\geq 0$, with equality if and only if $(M,g)=(\mathbb{R}^n,g_E)$.