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 boundary data are nonincreasing, the estimate yields rigidity, removing Case-Wang's pinching condition $\sup_\Sigma H_g\le 3\inf_\Sigma H_g$. For $n\ge 5$, the constant-data case also classifies the smooth critical metrics associated with their sharp $\sigma_2$ Sobolev trace conjecture. Second, within positive Einstein conformal classes, we extend the constant-$\sigma_2$ rigidity results of Viaclovsky and Gursky-Streets to nonincreasing prescribed data, including backgrounds with nonzero Weyl curvature for $n\ge 5$. Third, we extend Li-Li's spherical $\sigma_2/\sigma_1$ rigidity and Guan-Wang's sharp integral inequality to positive Einstein backgrounds, with the latter holding for $n\ge 5$ under positive scalar curvature. Fourth, we extend Frank-Peteranderl's spherical $\sigma_2$ stability to fixed nonround positive Einstein backgrounds in dimensions $n\ge 5$, retaining $H^1$ and $W^{1,4}$ control under positive scalar curvature.
Let $(v,p)$ be a smooth stationary Navier--Stokes flow in $\mathbb R^3$ that vanishes at infinity, and set $\omega :=\operatorname{curl}v$. We prove the endpoint implication \[ \omega\in L^{9/5,\infty}(\mathbb R^3) \quad\Longrightarrow\quad v\equiv0. \] This weak-Lorentz condition is invariant under the far-field rescaling associated with the borderline vorticity decay $|\omega(x)|=O(|x|^{-5/3})$ and strictly extends the $L^{9/5}$ vorticity criterion of Chae--Wolf. It also removes the relative smallness condition from the critical pointwise criterion of Kozono--Terasawa--Wakasugi: the decay $|\omega(x)|=O(|x|^{-5/3})$ alone implies $v\equiv0$. No finite Dirichlet energy is assumed. In our proof, we firstly use the endpoint Biot--Savart mapping and the critical annular Lorentz estimate of Seregin--Wang to yield finite Dirichlet energy. A logarithmic bound for the cumulative $L^{9/5}$ mass then selects blow-down scales whose stationary Euler limit satisfies Bernoulli companion laws on the exterior region. Together with the inherited weak endpoint bounds, these laws force the limiting energy flux to vanish. A harmonic-cutoff identity transfers this vanishing to the original scale and yields zero Dirichlet energy.
We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p<\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key independent ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Thus pointwise strict spacelikeness automatically improves to a uniform spacelike gap, including in the critical and supercritical regimes. The proof combines a geometric Bernstein estimate, comparison with an explicit hyperbolic cap, and weighted trace-free tensor identities. The result extends the known radial nonexistence theorem to arbitrary entire solutions and yields a geometric half-space rigidity theorem for complete spacelike hypersurfaces.
Xi-nan Ma, Tian Wu, Wangzhe Wu et al.· 0 citations
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