Let $(M^{n+1},g)$ be a compact Riemannian manifold with boundary. Under the assumptions $\Ric_g\geq ng$ and $\II_g\geq0$, Wang proposed a sharp strengthening of the Choi--Wang--Reilly estimate, asserting that the first nonzero Laplace eigenvalue of the boundary is at least $n$; see [J. Geom. Anal. 31 (2021)]. We disprove this assertion in every dimension $n+1\geq3$. More precisely, we construct a sequence of metrics on the hemisphere $\Sph^{n+1}_{+}$ converging in $C^\infty$ to the round metric and satisfying \[ \Ric_g>n g,\qquad \II_g>0,\qquad \lambda_1(\partial\Sph^{n+1}_{+},g|_{\partial\Sph^{n+1}_{+}})<n. \] The construction starts from Zhu's infinitesimal conformal deformation, which lowers one branch of the first boundary eigenspace while preserving the normalized Ricci lower bound to first order. We add a multiple of the spherical height function.
Escobar (J Funct Anal 165(1):101-116, 1999) conjectured that for every $n\ge 3$, an $n$-dimensional compact Riemannian manifold with nonnegative Ricci curvature and all boundary principal curvatures bounded below by $\kappa>0$ must satisfy $\sigma_1\geq \kappa$. We disprove this conjecture for every $n\geq 3$ by constructing conformal deformations of the Euclidean unit ball. We first establish a perturbative criterion, then construct explicit polynomial conformal factors satisfying this criterion. For every sufficiently small $t>0$, the resulting metrics $g_t=e^{2t\Phi}g_{\mathbb{R}^n}$ have positive Ricci curvature, every boundary principal curvature is strictly larger than $1$, and $\sigma_1(\mathbb{B}^n,g_t)<1$. The proof requires several computations, some of which were carried out in Mathematica. The Mathematica code is attached to this submission.
We give counterexamples to Yau's question on normalized scalar curvature integrals in every dimension $n\geq4$. For each such $n$, there exists a smooth complete Riemannian metric $g$ on $\mathbb{R}^n$ with nonnegative Ricci curvature and a pole such that \[ \lim_{R\to\infty}R^{2-n}\int_{B_q(R)}\mathrm{Scal}_g\,\mathrm dV_g=+\infty \] for every fixed $q\in\mathbb{R}^n$. Here $B_q(R)$ denotes the geodesic ball of radius $R$ centered at $q$, and $\mathrm{Scal}_g$ is the scalar curvature of $g$.
We study volume comparison for closed Riemannian manifolds satisfying a positive Ricci curvature lower bound together with an improved scalar curvature lower bound. We prove that if a closed Riemannian manifold $(N^n, g)$ satisfies $\operatorname{Ric}_g\ge (n-1)g$ and the scalar curvature $R_g\ge n(n-1)(1+\varepsilon)$, then its volume satisfies $$\lvert N\rvert_g \le \frac{1}{\sqrt{1+n\varepsilon}}\lvert\mathbb S^n\rvert. $$ In fact, assuming only $\mathrm{Ric}_g\ge(n-1)g$, we can prove that $$\frac{|N|_g}{\left|\mathbb{S}^n\right|} \le \frac{1}{|N|_g} \int_N\left(\frac{R_g}{n-1}-(n-1)\right)^{-\frac{1}{2}} d \mathrm{vol}_g. $$ The equality holds if and only if $N$ is isometric to the unit sphere. The proof combines a coefficient-adapted Jacobian comparison with a new integral shuffling comparison for scalar Jacobi solutions, inspired by Brown and Freedman \cite{BrownFreedman2022}. This yields an estimate that retains the full Ricci spectrum. The resulting volume bound agrees to first order in $\varepsilon$ with the factor appearing in Bray's conjecture. A further consequence of the argument is an averaged volume comparison for metric balls involving the scalar curvature. We also obtain a volume comparison theorem under a weighted integral lower bound on the Branson $Q$-curvature.
Let $(M^n, g, f)$ be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that (i) $Ric \geq \frac{\nabla_{\nabla f}Ric}{f}$ on $M\setminus D$, where $D$ is a compact set over $M$; (ii) $(M^n, g, f)$ smoothly converges to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$, we conclude that $(M^n, g, f)$ is isometric to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$. Notably, condition \textup{(i)} is weaker than the radial flatness condition in \cite{Petersen-Wylie2}.
Let $V$ be an $n$-dimensional Euclidean vector space, ,where $n\ge 4$, and $\ell = \lfloor\frac{n}{2}\rfloor$. We prove the sharp pointwise estimate \[ q_2(E) \ge -\frac{2(\ell -1)}{3\ell} \mathrm{Scal}(E) \mathrm{Id}_{\Lambda^2V^*} \] for every algebraic curvature tensor $E$ on $V$ with nonnegative sectional curvature. Applying this estimate to the decomposition $\operatorname{Rm}_{g}=K_{\min}I+E$, we obtain the vanishing of $H^2(M; \mathbb{R})$ under a dimension-dependent strict sectional-scalar curvature pinching condition. At the weak endpoint, all harmonic two-forms are parallel. Apart from the flat case, this yields $b_2(M)=0$ in odd dimensions and $b_2(M)\le 1$ in even dimensions. At even-dimensional endpoint, $b_2(M)>0$ forces $(M, g)$ to be isometric, up to scaling, to $\mathbb{CP}^{\ell}$ with its Fubini-Study metric. As a consequence every closed five-dimensional manifold satisfying the strict pinching condition implies is a rational homology sphere. An anisotropic rescaling of the same homogeneous four-frame estimate also gives the sharp pointwise sectional-scalar pinching criterion \[ K_{\min} \ge \frac{n(n-1)}{n^2-n+12}S_0 \quad \Longrightarrow \mathrm{PIC2}. \] The strict pinching places the curvature tensor in the interior of PIC2 and normalized Ricci flow brings it to a positive constant sectional curvature.
We establish sharp Sobolev-type geometric inequalities on $\mathbb{S}^n$ involving the total $\sigma_k$-curvatures $\int_{\mathbb{S}^n}\sigma_k(g)\,dv_g$. These results extend the optimal inequalities of Guan--Wang~\cite{GWDuke} from the cone $\mathcal{C}_k$ to the strictly larger cone $\mathcal{C}_{k-1}$, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in $\mathcal{C}_{k-1}$ introduces substantial analytic difficulties; in particular, one must obtain $C^2$ a priori estimates while simultaneously verifying that the flow remains parabolic. We resolve these issues via a carefully designed test function and by applying the maximum principle to the maximal eigenvalue of the Hessian matrix. As applications, we solve two open problems in dimensions 3 and 4. Finally, we give examples to show that these inequalities cannot be extended to $\mathcal{C}_{k-2}$.
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