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Jikang Wang

Publications and source records attributed to Jikang Wang.

8 recordsLinked to original sources
arXiv2026

Fibrations, the First Betti Number, and Almost Nonnegative Ricci Curvature

In this paper, we prove fibration theorems for manifolds with almost nonnegative Ricci curvature and certain extra regularity assumptions. We show that a closed $n$-manifold $M$ satisfying $\mathrm{diam}(M)^2\mathrm{sec}_M \geq -κ$ and $\mathrm{diam}(M)^2\mathrm{Ric}_M \geq -δ$, where $δ>0$ is sufficiently small depending only on $n$ and $κ$, fibers over a $b_1(M)$-torus. This removes the upper sectional curvature bound required in the earlier result of Yamaguchi \cite{Y88}. As a corollary, we obtain a refinement of Yamaguchi's smooth fibration theorem (\cite{Y91}), showing that the fiber itself (rather than a finite cover of it) fibers over a $b_1$-torus. Our results extend to manifolds satisfying a generalized Reifenberg condition introduced in \cite{HH24}, which encompasses both a lower bound on sectional curvature and the local rewinding Reifenberg condition. In the nonsmooth setting, a similar result also holds for a non-collapsed $\mathrm{RCD}(-ε(D,r,n),n)$ space whose diameter is bounded by $D$ and which satisfies the $(r,δ(n))$-local rewinding Reifenberg condition. The proofs rely on an equivariant regularity theorem for almost submetries under a lower Ricci curvature bound. In addition, we study the stability of rank of Abelian actions along equivariant Gromov-Hausdorff convergence in this paper.

math.DG↗
arXiv2024

On the limit of simply connected manifolds with discrete isometric cocompact group actions

We study complete, connected and simply connected $n$-dim Riemannian manifold $M$ satisfying Ricci curvature lower bound. Further more, suppose that $M$ admits discrete isometric group actions $G$ so that the diameter of the quotient space $\mathrm{diam}(M/G)$ is bounded. In particular, for any $n$-manifold $N$ satisfying $\mathrm{diam}(N) \le D$ and $\mathrm{Ric} \ge -(n-1)$, the universal cover and fundamental group $(\widetilde{N},G)$ satisfies the above condition. Let $\{(M_i,p_i)\}_{i \in \mathbb{N}}$ be a sequence of complete, connected and simply connected $n$-dim Riemmannian manifolds satisfying $\mathrm{Ric} \ge -(n-1)$. Let $G_i$ be a discrete subgroup of $\mathrm{Iso}(M_i)$ such that $\mathrm{diam}(M_i/G_i) \le D$ where $D>0$ is fixed. Passing to a subsequence, $(M_i, p_i,G_i)$ equivariantly pointed-Gromov-Hausdorff converges to $(X,p,G)$. Then $G$ is a Lie group by Cheeger-Colding and Colding-Naber. We shall show that the identity component $G_0$ is a nilpotent Lie group. Therefore there is a maximal torus $T^k$ in $G$. Our main result is that $X/T^k$ is simply connected. Moreover, $π_1(X,p)$ is generated by loops contained in the $T^k$-orbits up to conjugation; each of these loops can be represented by $α^{-1} \cdot β\cdot α$ where $α$ is a curve from $y$ to $p$ for some $y \in X$, and $β$ is a loop at $y$ contained in the $T^k$-orbit of $y$.

math.DG↗
arXiv2024

On the non-collapsed RCD spaces with local bounded covering geometry

We consider a RCD$(-(N-1),N)$ space $(X,d,\mathcal{H}^N)$ with local bounded covering geometry. The first result is related to Gromov's almost flat manifold theorem. Specifically, if for every point $\tilde{p}$ in the universal cover $\widetilde{X}$, we have $\mathcal{H}^N(B_1(\tilde{p})) \ge v > 0$ and the diameter of $X$ is sufficiently small, then $X$ is biHölder homeomorphic to an infranil-manifold. Moreover, if $X$ is a smooth Riemannian $N$-manifold with $\mathrm{Ric} \ge -(N-1)$, then $X$ is biHölder diffeomorphic to an infranil-manifold. An application of our argument is to confirm the conjecture that Gromov's almost flat manifold theorem holds in the $\mathrm{RCD}+\mathrm{CBA}$ setting. The second result concerns a regular fibration theorem. Let $(X_i,d_i,\mathcal{H}^N)$ be a sequence of RCD$(-(N-1),N)$ spaces converging to a compact smooth $k$-dimensional manifold $K$ in the Gromov-Hausdorff sense. Assume that for any $p_i \in X_i$, the local universal cover is non-collapsing, i.e., for any pre-image point $\tilde{p}_i$ of $p_i$ in the universal cover of the ball $B_3(p_i)$, we have $\mathcal{H}^N(B_{1}(\tilde{p}_i)) \ge v$ for some fixed $v>0$. Then for sufficiently large $i$, there exists a fibration map $f_i:X_i \to K$, where the fiber is an infra-nilmanifold and the structure group is affine.

math.DG↗
arXiv2022

$\mathrm{RCD}^{*}(K,N)$ spaces are semi-locally simply connected

It was shown by Mondino-Wei that any $\mathrm{RCD}^{*}(K,N)$ space $(X,d,\mathfrak{m})$ has a universal cover. We prove that for any point $x \in X$ and $R>0$, there exists $r<R$ such that any loop in $B_r(x)$ is contractible in $B_R(x)$; in particular, $X$ is semi-locally simply connected and the universal cover of $X$ is simply connected. This generalizes the author's earlier work that any Ricci limit space is semi-locally simply connected.

math.DG↗
arXiv2021

Ricci Limit Spaces Are Semi-locally Simply Connected

Let $(X,p)$ be a Ricci limit space. We show that for any $ε> 0$ and $x \in X$, there exists $r< ε$, depending on $ε$ and $x$, so that any loop in $B_{r}(x)$ is contractible in $B_ε(x)$. In particular, $X$ is semi-locally simply connected. Then we show that the generalized Margulis lemma holds for Ricci limit spaces of $n$-manifolds.

math.DG↗
arXiv2021

Some topological results of Ricci limit spaces

We study the topology of a Ricci limit space $(X,p)$, which is the Gromov-Hausdorff limit of a sequence of complete $n$-manifolds $(M_i, p_i)$ with $\mathrm{Ric}\ge -(n-1)$. Our first result shows that, if $M_i$ has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then $X$ is semi-locally simply connected. In the process, we establish a slice theorem for isometric pseudo-group actions on a closed ball in the Ricci limit space. In the second result, we give a description of the universal cover of $X$ if $M_i$ has a uniform diameter bound.

math.DG↗
arXiv2018

A model of orbital angular momentum Li-Fi

Twisted light has recently gained enormous interest in communication systems. Thus far, twisted light has not yet been utilized for visible light communication to transmit data. Here, by exploiting the color and orbital angular momentum (OAM) degrees of freedom simultaneously, we construct a much higher-dimensional space spanned by their hybrid mode basis, which further increases the information capacity of twisted light. We build a new visible light communication system using a white light emitting diode, with red, green and blue (RGB) colors serving as independent channels and with OAM superposition states encoding the information. We connect our conceptually new RGB-OAM hybrid coding with the specially designed two-dimensional holographic gratings based on theta-modulation. After indoor free-space transmission, we decode the color information with an Xcube prism and subsequently decode the OAM superposition states with a pattern recognition method based on supervised machine learning. We succeed in demonstrating the transmission of color images and a piece of audio with the fidelity over 96%. Our point-to-point scheme with hybrid RGB-OAM encoding, not only increases significantly the information capacity of twisted light, but also offers additional security that supplements the traditional broadcasting visible light communications, e.g., Li-Fi.

physics.app-ph↗