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 nonnegative scalar curvature and constant $I_a$ is Einstein whenever $a\ge-4$. The scalar-curvature sign assumption can be removed for $-4\le a\le-2(n-2)/(n-1)$. We also extend the quotient rigidity theorem of Ge--Wang--Wei to $I_a=\Lambda R^\theta$ for $R>0$, $a\ge-4$, and $\theta\le1$. Sharpness is established in every dimension by a shooting construction. For each sufficiently small $\eta>0$, we construct a smooth non-Einstein metric in the round conformal class on $\mathbb S^n$ with positive scalar curvature and the same constant $I_{-4-\eta}$ as the unit round metric. The construction smoothly matches a perturbed Schwarzschild neck near the equator with perturbed round caps at both poles.
Using a reference-curvature technique, we prove four main theorems. (i) For $n\geq4$, we remove the mean-curvature pinching from the Case-Wang rigidity theorem for $\sigma_2$-flat conformal hemispheres with $A_g\in\overline{\Gamma_2^+}$ and positive constant natural boundary curvature. (ii) On every closed positive Ein...
We study nonnegative entire spacelike graphs in Lorentz--Minkowski space satisfying $\sigma_k(A[u])=u^p$. For $2\le k<n$ and $p\ge k$, pointwise strict spacelikeness and closed $k$-admissibility yield universal bounds for the height and Lorentz factor. Their proof combines block coercivity in the full G\r{a}rding cone,...
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...
We establish a quantitative stability estimate for codimension-one immersions into a round sphere. Let $(M,g)$ be an oriented Riemannian manifold and suppose that a prescribed shape operator is realised by a smooth isometric immersion $\theta:(M,g)\to\mathbb S^{n+1}$. We prove that, on every relatively compact strongly...
We prove that every closed $m$-quasi-Einstein manifold $(M^n,g,X,\lambda)$ with constant $\lambda\leq0$ is trivial whenever $m\leq-2$. This establishes the Colling--Dunajski conjecture in the range $m\leq-2$ and, for $n\geq5$, extends the previously known range $m\leq2-n$. This result is placed within a broader study o...
For every pair of integers $3\leq k<n$ with $k$ odd, we construct a compact embedded rotational torus in $\mathbb{R}^{n+1}$ whose homothetic dilations satisfy the unnormalised $\sigma_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has H\"older regularity $C^{1,1/k}$ and Sobolev regularity $W^...
