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Zhu Ye

Publications and source records attributed to Zhu Ye.

5 recordsLinked to original sources
arXiv2025

Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature

Let $M$ be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. In this paper, we study whether the volume growth order of $M$ is always greater than or equal to the dimension of some (or every) asymptotic cone of $M$. Our first main result asserts that, under the conic at infinity condition, if the infimum of the volume growth order of $M$ equals $k$, then there exists an asymptotic cone of $M$ whose upper box dimension is at most $k$. In particular, this yields a complete affirmative answer to our problem in the setting of nonnegative sectional curvature. In the subsequent part of the paper, we extend or partially extend Sormani's results concerning $M$ with linear volume growth to more relaxed volume growth conditions. Our approach also allows us to present a new proof of Sormani's sublinear diameter growth theorem for open manifolds with $\mathrm{Ric}\geq 0$ and linear volume growth. Finally, we construct an example of an open $n$-manifold $M$ with $\mathrm{sec}_M\geq0$ whose volume growth order oscillates between 1 and $n$.

math.DG↗
arXiv2024

On manifolds with nonnegative Ricci curvature and the infimum of volume growth order $<2$

We prove two rigidity theorems for open (complete and noncompact) $n$-manifolds $M$ with nonnegative Ricci curvature and the infimum of volume growth order $<2$. The first theorem asserts that the Riemannian universal cover of $M$ has Euclidean volume growth if and only if $M$ is flat with an $n-1$ dimensional soul. The second theorem asserts that there exists a nonconstant linear growth harmonic function on $M$ if and only if $M$ is isometric to the metric product $\mathbb{R}\times N$ for some compact manifold $N$.

math.DG↗
arXiv2023

Maximal first Betti number rigidity of noncompact $\texttt{RCD}(0,N)$ spaces

Let $(M,d,\mathfrak{m})$ be a noncompact $\texttt{RCD}(0,N)$ space with $N\in\mathbb{N}_+$ and $\text{supp}\mathfrak{m}=M$. We prove that if the first Betti number of $M$ equals $N-1$, then $(M,d,\mathfrak{m})$ is either a flat Riemannian $N$-manifold with a soul $T^{N-1}$ or the metric product $[0,\infty)\times T^{N-1}$, both with the measure a multiple of the Riemannian volume, where $T^{N-1}$ is a flat torus.

math.DG↗
arXiv2018

Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then $\pi_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\widetilde{M}$ has Euclidean volume growth of constant at least $L$, then we can bound the index of that abelian subgroup in terms of $n$ and $L$. In particular, our result implies that if $\widetilde{M}$ has Euclidean volume growth of constant at least $1-\epsilon(n)$, then $\pi_1(M)$ is finitely generated and $C(n)$-abelian.

math.DG↗