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 for $a\ge-4$. This lower threshold is sharp in every dimension: for each sufficiently small $\eta>0$, the round sphere admits a smooth non-Einstein conformal metric with positive scalar curvature and constant $I_{-4-\eta}(g)$. The rigidity proof combines the pointwise Obata identity with a Newton identity whose reference curvature is the minimum of the scalar curvature. A local pole equation and integral estimates yield a radial shooting construction joining a neck to nearly round caps. The rescaled necks converge to the Riemannian Schwarzschild metric, and we determine the asymptotic neck scale. We also prove rigidity for $I_a(g)=\Lambda R_g^\theta$ when $R_g>0$, $a\ge-4$, and $\theta\le1$. In dimension four, constant-$I_a(g)$ rigidity holds for $-4\le a\le-4/3$ without a scalar-curvature sign assumption.
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-minimizin...
Let $(M^n,g_0)$ be a closed hyperbolic manifold, $n\geq3$. We compute the Hessian of volume entropy at $g_0$ under the constraint $Sc_g=-n(n-1)$ and express it as a quadratic form on transverse-traceless symmetric two-tensors. In dimension three, the Hessian is negative definite, and every sufficiently $C^\infty$-close...
We completely classify conformal metrics on the unit ball $(\mathbb{B}^{n+1},|\mathrm{d} x|^2)$, $n\geq4$, with positive constant $Q$-curvature, positive constant $T$-curvature, and minimal boundary. After normalizing the $Q$-curvature, there is a unique conformal metric for each $T$-curvature value in $[0,+\infty)$, u...
Using classical conformal divergence identities with a variable reference curvature, we prove four main results. First, we establish a general mean-curvature estimate for conformal metrics on the round hemisphere with $A_g\in\overline{\Gamma_2^+}$ and positive prescribed $H_2$ data. When $\sigma_2(A_g)=0$ and the bound...
Let $2\leq k<n/2$ and let $(M^n,[g])$ be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a $k$-admissible metric in the conformal class $[g]$. We prove a stability result of the $\sigma_k$-curvature inequality on $M$, in the sense that if equality is almost satisfied for some conforma...
Let $(M,g)$ be a compact oriented Einstein four-manifold with Einstein constant $E$ and let $\widehat{R}^+$ denote the action of the Riemann curvature tensor on self-dual two-forms. We show that if $\widehat{R}^+<0$, then $g$ is strictly linearly stable for the Einstein-Hilbert functional, thus giving a chiral criterio...
Diego Artacho· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
