Abstract:Let \(g\) be a smooth Riemannian metric and $H$ a function on $\mathbb{S}^2$ and let $\lambda_1(g)$ be the first eigenvalue of the operator $(-\Delta_g+K_g)$. We prove that the Bartnik mass of the triple $(\mathbb{S}^2,g,H)$ is bounded above by $\sqrt{|\mathbb{S}^2|_g/16\pi}$ provided either (a) $\lambda_1(g)=0$ and $H>0$; or (b) $\lambda_1(g)>0$ and $H\geq 0$. The eigenvalue condition, in particular, imposes no lower bound on $K_g$ (even under an area constraint) and thereby extends previous results which assume $K_g\geq 0$.
From: Albert Chau [view email]
[v1]
Sat, 20 Jun 2026 00:41:15 UTC (9 KB)
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