Aug 2026· 3 citations· ⚡ 2 influential· 48 references
Mathematics
Abstract
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 growth of order $R^{m-1}$ for $R\le k^{-1/2}$. At larger scales we obtain the corresponding estimate with an exponential factor $\exp(C\sqrt k R)$, and an example shows that this factor is necessary.
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 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 wit...
Let $(M,g)$ be a connected, compact, $n$-dimensional Riemannian manifold with $\operatorname{Ric}(M,g)\geq-(n-1)\kappa g$. We introduce a weighted combinatorial Laplacian on $\varepsilon$-discretizations of $M$ and prove a spectral comparison theorem between the weighted graph Laplacian and the Laplace-Beltrami operato...
Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the vo...
For every $n\ge4$ and $L>0$, we construct a smooth $4$-PIC metric on $S^n$ with Urysohn $1$-width at least $L$ and an embedded stable minimal disk of intrinsic inradius at least $L$. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed...
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),...
Yu-Ting Wu· 0 citations
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