Let $(M^3,g)$ be a complete Riemannian manifold diffeomorphic to $\R^3\setminus\{0\}$, with nonnegative scalar curvature. Assume that a distinguished end is asymptotically flat, with ADM mass $m_+$. For each $p\in(1,3)$, define $c_{O,p}$ as the infimum of the Schwarzschild-normalized $p$-capacity over outward-minimizing finite-perimeter boundaries separating the two ends. We prove that $m_+\ge c_{O,p}$ whenever $A(g)>0$, where $A(g)$ is the infimum of the areas of boundaries separating the two ends. Equality at a single exponent produces a least-area horizon, forces its exterior to be the Schwarzschild exterior of mass $m_+$, and yields equality at every exponent. In the equality case, if the second end is also asymptotically flat, its mass satisfies $m_-\ge m_+$, with equality precisely for the two-sided spatial Schwarzschild manifold. We also show that a strict gap between $c_{O,p}$ and the unconstrained capacity infimum $c_{M,p}$ detects a horizon.
Let $(M^{n}, g)$ be a complete, simply connected Riemannian manifold without boundary, of dimension $n\ge3$, with curvature operator at least that of the unit sphere. We prove that $$\int_M {\rm scal}(x)\ d {\rm Vol}_x\le n(n-1)\omega_n,$$ where $\omega_n$ is the volume of the unit $n$-sphere. Equality holds if and onl...
Let $I_a(g)=Q_g+a\sigma_2(A_g)$, where $A_g$ is the Schouten tensor and $Q_g$ is Branson's $Q$-curvature. On a closed connected manifold of dimension $n\ge4$ with a positive Einstein metric $g_0$, we prove that every smooth metric conformal to $g_0$ with nonnegative scalar curvature and constant $I_a(g)$ is Einstein fo...
Let $M^{n+1}$ be a closed, connected smooth manifold, where $n\geq 1$. For each prescribed volume fraction $s\in(0,1)\setminus\{\frac12\}$, we prove that the isoperimetric region of volume $s \operatorname{Vol}_g(M)$ is unique for a generic set of smooth Riemannian metrics $g$. At half volume, a generic metric has exac...
Let $S\subset\mathbb R^3$ be a properly embedded mean-convex planar surface with finitely many ends. Designate one end as asymptotically flat, assume that $H_S$ is integrable there, and denote its extrinsic mass by $m_+(S)$. Let $A_S$ be the infimum of the areas of compact surfaces separating the distinguished end from...
Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective...
Ren-Yu Chen, Song-Ying Li, Su-Juan Long et al.· 0 citations
For every dimension $m\geq 3$, we show the existence of an open set $U\subset \mathbb R^{m+1}$, a smooth Riemannian metric $g$ on it, and an integral $m$-dimensional stationary varifold $V$ in $(U,g)$ with the following properties. $V$ has a flat tangent plane of multiplicity $2$ at an interior point $p$, it is a smoot...
Camillo De Lellis, Jonas Hirsch, Zachary Lihn et al.· 0 citations
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