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We resolve Case's question (Crelle's Journal, 2024) on the sharp lower threshold for Obata-type rigidity of $I_a=Q+a\sigma_2(A)$ in positive Einstein conformal classes of dimension $n\ge4$. Using a new reference-curvature identity, we prove that, on a closed connected manifold, every smooth metric in such a class with...
Let $(v,p)$ be a smooth stationary Navier--Stokes solution in $\mathbb R^3$ with $v(x)\to0$ as $|x|\to\infty$. We prove that $v\equiv0$ if its vorticity $\omega:=\nabla \times v$ belongs to $L^{s,\infty}(\mathbb R^3)$ for some $9/5\le s\le1.87$. This gives a global weak-Lorentz vorticity criterion beyond $9/5$ requirin...
Let $(M^n,g_0)$ be a closed connected Einstein manifold with positive scalar curvature, where $n\ge6$. Using an Obata-type identity, we prove that every smooth metric conformal to $g_0$ with constant sixth-order $Q$-curvature is Einstein.
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 $1\leqslant p\leqslant\frac{n+2}{n-2}$, every nonnegative $C^2$ solution...
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...
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 $(v,p)$ 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 resca...
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 satisfyi...
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