Let $(M^n,g)$ be complete, let $u>0$, and assume $\operatorname{Ric}_g-\alpha\frac{\Delta_g u}{u}g\ge (n-1)\kappa g$. We introduce mixed radial balls associated with the conformal metric $u^{2\alpha}g$. For these mixed radial balls, we obtain a space-form-sharp model comparison and polynomial weighted volume growth without pointwise bounds on $u$. As applications, we give a radial derivation of the spectral Bonnet--Myers and sharp volume theorem of Antonelli--Xu, and a short volume growth derivation of the stable Bernstein theorem in $\mathbb{R}^4$.
Let $(M, g)$ be an $n$ $(\ge 3)$ dimensional, non-collapsed compact Riemannian manifold and $\operatorname{Ric}^-$ be the negative part of the Ricci curvature and $\beta \in (\frac{2n}{n+2}, 2)$. We prove a relative volume comparison theorem when $|\operatorname{Ric}^-|^\beta$ is in the Kato class (cf. Definition 1.1),...
Let $(M^n,g)$ be a complete Riemannian manifold with $\Ric\ge-kg$ and uniformly positive $m$-intermediate curvature in the sense of Brendle--Hirsch--Johne. We prove that the Fisher eigenvalue $\lambda_{n-m+1}$ is small, after heat averaging, below the curvature scale $k^{-1}$. Consequently, balls have polynomial volume...
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 $(M^{n+1},g)$ 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 dispro...
We study the central Hardy--Littlewood maximal operator $M_c$ in $\mathbb{R}^n$ for all $n\geq1$, with particular emphasis on the question of its injectivity. More precisely, we consider whether there exist two nonnegative functions $f,g\in L^1(\mathbb{R}^n)$ such that $\|f-g\|_1>0$ and $M_cf=M_cg$. Along the way, we o...
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...
Hao-Xuan Cheng· 2 citations
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