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 $g$.
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...
Let $(M^n,g)$ be a compact Riemannian manifold of dimension $n\ge 2$, possibly with nonempty convex $C^2$ boundary. We study the first nonzero eigenvalue $\lambda_1(M)$ of the Laplace--Beltrami operator, subject to the Neumann boundary condition when $\partial M\neq\emptyset$. Given a reference scalar curvature $R_E>0$...
Let $(M^n,g)$ be a complete connected noncompact smooth Riemannian manifold without boundary, with $n\ge3$ and positive scalar curvature. We prove that $M$ admits a complete smooth metric of uniformly positive scalar curvature if $$ \liminf_{d_g(o,x)\to\infty}d_g(o,x)^2R_g(x)>\frac{n-1}{n} $$ for a fixed point $o\in M$...
Let $M^n$, $n\ge3$, be the smooth, connected, complete, noncompact boundary of a closed convex set with nonempty interior in $\mathbb{R}^{n+1}$. We prove that its normalized scalar-curvature integral at infinity is at most $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$, except when, up to an ambient rigid motion, $M$ is t...
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 $R_g\ge n(n-1)(1+\varepsilon)$...
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 t...
