A New Proof of the Gromov's Theorem on Almost Flat Manifolds

Gromov-Hausdorff和内在平坦收敛的RCD(K,N)和Kato空间

Gromov-Hausdorff and intrinsic flat convergence of RCD(K,N) and Kato spaces

本文研究了RCD(K,N)和Kato空间在Gromov-Hausdorff和内在平坦收敛下的定向性稳定性及极限一致性。

Fano manifolds with weak almost Kähler-Ricci solitons

In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded below, there exists a sequence of weak almost Kähler-Ricci solitons which converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology.

Mixed curvature almost flat manifolds

We prove a mixed curvature analogue of Gromov's almost flat manifolds theorem for upper sectional and lower Bakry-Emery Ricci curvature bounds.

Intrinsic flat convergence of points and applications to stability of the positive mass theorem

We prove results on intrinsic flat convergence of points---a concept first explored by Sormani in \cite{Sormani-AA}. In particular, we discuss compatibility with Gromov-Hausdorff convergence of points---a concept first described by Gromov in \cite{Gromov-poly}. We apply these results to the problem of stability of the positive mass theorem in mathematical relativity. Specifically, we revisit the article \cite{HLS} on intrinsic flat stability for the case of graphical hypersurfaces of Euclidean space: We are able to fill in some details in the proofs of Theorems 1.4 and Lemma~5.1 of \cite{HLS} …

在 Ricci 和积分标量曲率界下环面的 Gromov-Hausdorff 稳定性

Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds

本文提出到平坦环面的非线性调和分裂映射，证明其存在性与 Ricci 及积分标量曲率界下的 Gromov-Hausdorff 靠近性等价，并结合 Stern 不等式建立了平坦 3-环面的新稳定性定理。

非等变格罗莫夫环面的存在性

The Existence of Non-Equivariant Gromov Tori

本文针对n≥3时平坦环面嵌入球面的问题，证明存在非等变嵌入格罗莫夫环面，给出该问题的否定答案，推进了最优曲率界相关研究。

Geometric Structures of Collapsing Riemannian Manifolds II: N*-bundles and Almost Ricci Flat Spaces

In this paper we study collapsing sequences M_{i}-> X of Riemannian manifolds with curvature bounded or bounded away from a controlled subset. We introduce a structure over X which in an appropriate sense is dual to the N-structure of Cheeger, Fukaya and Gromov. As opposed to the N-structure, which live over the M_{i} themselves, this structure lives over X and allows for a convenient notion of global convergence as well as the appropriate background structure for doing analysis on X. This structure is new even in the case of uniformly bounded curvature and as an application we give a generali…

Almost maximal volume entropy rigidity for integral Ricci curvature in the non-collapsing case

In this note we will show the almost maximal volume entropy rigidity for manifolds with lower integral Ricci curvature bound in the non-collapsing case: Given $n, d, p>\frac{n}{2}$, there exist $δ(n, d, p), ε(n, d, p)>0$, such that for $δ<δ(n, d, p)$, $ε<ε(n, d, p)$, if a compact $n$-manifold $M$ satisfies that the integral Ricci curvature has lower bound $\bar k(-1, p)\leq δ$, the diameter $diam(M)\leq d$ and volume entropy $h(M)\geq n-1-ε$, then the universal cover of $M$ is Gromov-Hausdorff close to a hyperbolic space form $\Bbb H^k$, $k\leq n$; If in addition the volume of $M$, $vol(M)\geq…

The collapsing geometry of almost Ricci-flat 4-manifolds

We consider Riemannian 4-manifolds that Gromov-Hausdorff converge to a lower dimensional limit space, with the Ricci tensor going to zero. Among other things, we show that if the limit space is two dimensional then under some mild assumptions, the limiting four dimensional geometry away from the curvature blowup region is semiflat Kaehler.

具有有界Ricci曲率的非坍缩流形的正交标架丛的Gromov-Hausdorff极限

Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature

研究具有双边有界Ricci曲率的非坍缩流形的正交标架丛在几乎标准度量下的Gromov-Hausdorff极限空间，证明其与Ricci极限空间类似，奇异集余维数≥4且补集为开稠密的C^{1,α}黎曼流形。

Conformal tori with almost non-negative scalar curvature

In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with uniformly controlled geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, $L^{p}$ sense and the measured Gromov-Hausdorff sense.

Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below

We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson-Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show that, under lower Ricci curvature bounds, almost Euclidean balls in Kahler manifolds admit good holomorphic coordinates. Further applications are integral bounds for the scalar curvature on balls, a…

稳定性与等变Gromov--Hausdorff收敛

Stability and equivariant Gromov--Hausdorff convergence

本文研究等变Gromov--Hausdorff收敛的应用，在紧有限维Alexandrov空间下证明稳定性结果，还提出几乎交换图概念以简化相关阐述与论证。

Cheeger-Gromov convergence in a conformal setting

For a sequence $\{(M_i, g_i, x_i)\}$ of pointed Riemannian manifolds with boundary, the sequence $\{(M_i,\tilde g_i,x_i)\}$ is its conformal satellite if the metric $\tilde g_i$ is conformal to $g_i$, that is, $\tilde g_i=u^{\frac{4}{n-2}}_ig_i$. Assuming the manifolds $(M_i,g_i,x_i)$ have uniformly bounded geometry, we show that both sequences have smoothly Cheeger-Gromov convergent subsequences provided the conformal factors $u_i$ are principal eigenfunctions of an appropriate elliptic operator. Part of our result is a Cheeger-Gromov compactness for manifolds with boundary. We use stable ver…

A four dimensional example of Ricci-flat metric admitting almost-Kähler non-Káhler structure

We construct an example of Ricci-flat almost-Kähler non-Kähler structure in four dimensions.

On three dimensional conformally flat almost cosymplectic manifolds

In the paper there are described new examples of conformally flat three dimensional almost cosymplectic manifolds. All these manifolds form a class which was completely characterized.

Properties of the Intrinsic Flat Distance

Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorff and intrinsic flat compactness theorems including the Tetrahedral Compactness Theorem. Much of the work in this paper builds upon Ambrosio-Kirchheim's Slicing Theorem combined with an adapted version Gromov's Filling Volume.

Relative Ruan and Gromov-Taubes Invariants of Symplectic 4-Manifolds

We define relative Ruan invariants that count embedded connected symplectic submanifolds which contact a fixed stable symplectic hypersurface V in a symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders (in addition to insertions on X\V) for stable V. We obtain invariants of the deformation class of (X,V,w). Two large issues must be tackled to define such invariants: (1) Curves lying in the hypersurface V and (2) genericity results for almost complex structures constrained to make V pseudo-holomorphic (or almost complex). Moreover, these invariants are refined to take…

On 4-dimensional, conformally flat, almost $ε$-Kählerian manifolds

The local structure of 4-dimensional, conformally flat, almost $ε$-Kählerian (i.e., almost pseudo-Kählerian and almost para-Kählerian) manifolds is characterized with the help of left-regular and right-regular paraquaternionic functions. Examples of such structures are discussed.

Metric rigidity of Kahler manifolds with lower Ricci bounds and almost maximal volume

In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of Kewei Zhang and Yuchen Liu on holomorphic rigidity of such Kahler manifolds with the structure theorem of Tian-Wang for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume

Scalar Curvature and Intrinsic Flat Convergence

Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic Flat convergence, developed jointly with Wenger. This notion has been applied successfully to study sequences that arise in General Relativity. Gromov has suggested it should be applied in other settings as well. We f…

On noncollapsed almost Ricci-flat 4-manifolds

We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the fundamental group is infinite or, more generally, of sufficiently large cardinality.

Small fiberwise oscillation of the eigenfunctions of collapsing Einstein manifolds

By Cheeger-Colding's almost splitting theorem, if a domain in a Ricci flat manifold is pointed-Gromov-Hausdorff close to a lower dimensional Euclidean domain, then there is a harmonic almost splitting map. We show that any eigenfunction of the Laplace operator is almost constant along the fibers of the almost splitting map, in the $L^2$-average sense. This generalizes an estimate of Fukaya in the case of collapsing with bounded diameter and sectional curvature.

Intrinsic flat convergence with bounded Ricci curvature

In this paper we address the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In \cite{SormaniWenger2010, SormaniWenger2011}, Sormani-Wenger show that for a sequence of Riemannian manifolds with nonnegative Ricci curvature, a uniform upper bound on diameter, and non-collapsed volume, the intrinsic flat limit exists and agrees with the Gromov-Hausdorff limit. This can be viewed as a non-cancellation theorem showing that for such sequences, points don't cancel each other out in the limit. Here we prove a similar no-cancellation theorem, rep…

Stability of graphical tori with almost nonnegative scalar curvature

By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show t…

On Bochner flat almost Kahler manifolds

The main purpose of this article is to prove that there exist no proper $AK_3$-manifold of dimension $2n\ge 6$ with vanishing Tricerri-Vanhecke Bochner curvature tensor and constant scalar curvature.

Half conformally flat gradient Ricci almost solitons

The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a steady traceless $κ$-Einstein soliton and is realized on the cotangent bundle of an affine surface.

Bach-flat h-almost gradient Ricci solitons

On an $n$-dimensional complete manifold $M$, consider an $h$-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and $dh/du>0$, then the manifold $M$ is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. Moreover, if the dimension of $M$ is four, the metric $g$ is conformally flat.

The Tetrahedral Property and a new Gromov-Hausdorff Compactness Theorem

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the same dimension. The tetrahedral property depends only on distances between points in spheres, yet we show it provides a lower bound on the volumes of balls. The proof is based upon intrinsic flat convergence and a new notion called the sliced filling volume of a ball.

Degenerations of Ricci-flat Calabi-Yau manifolds

This paper is a sequel to arXiv:1012.2940. We further investigate the Gromov-Hausdorff convergence of Ricci-flat Kähler metrics under degenerations of Calabi-Yau manifolds. We extend Theorem 1.1 in arXiv:1012.2940 by removing the condition on existence of crepant resolutions for Calabi-Yau varieties.

On $L^{2}$-harmonic forms of complete almost Kähler manifold

In this article, we study the $L^{2}$-harmonic forms on the complete $2n$-dimensional almost Käher manifold $X$. We observe that the $L^{2}$-harmonic forms can decomposition into Lefschetz powers of primitive forms. Therefore we can extend vanishing theorems of $d$(bounded) (resp. $d$(sublinear)) Kähler manifold proved by Gromov (resp. Cao-Xavier, Jost-Zuo) to almost Kählerian case, that is, the spaces of all harmonic $(p,q)$-forms on $X$ vanishing unless $p+q=n$. We also give a lower bound on the spectra of the Laplace operator to sharpen the Lefschetz vanishing theorem on $d$(bounded) case.

Ricci-平坦Kähler 4-流形基本群的不稳定性

Instability of the Fundamental Group for Ricci-Flat Kähler 4-Manifolds

本文证明在非坍缩Gromov–Hausdorff收敛下，闭Ricci-平坦Kähler四流形的基本群不局部稳定，并构造收敛到同一紧致平坦轨道流形的具体族。

Fibrations, and stability for compact group actions on manifolds with local bounded Ricci covering geometry

In this work, we (partially) generalize two classical tools in study of collapsed manifolds with bounded sectional curvature: a (singular) fibration theorem by Fukaya (1987) and Cheeger-Fukaya-Gromov (1992), and the stability for isometric compact Lie group actions on manifolds by Palais (1961) and Grove-Karcher (1973), to manifolds with local bounded Ricci covering geometry. Our two generalized results have been used in a recent work of Xiaochun Rong in generalizing Gromov's almost flat manifolds theorem to maximally collapsed manifolds with local bounded Ricci covering geometry.

Bergman Kernels and algebraic structure of limit space for a sequence of almost Kähler-Ricci solitons

In this paper, we give a lower bound of Bergman kernels for a sequence of almost Kähler-Einstein Fano manifolds, or more general, a sequence of Fano manifolds with almost Kähler-Ricci solitons. This generalizes a result by Donaldson-Sun, Tian for Kähler-Einstein manifolds sequence with positive scalar curvature. As an application of our result, we prove that the Gromov-Hausdorff limit of sequence is homomorphic to a log terminal $Q$-Fano variety which admits a Kähler-Ricci soliton on its smooth part.

几乎Ricci平坦5-流形的Kummer型构造

Kummer-type constructions of almost Ricci-flat 5-manifolds

本文利用Kummer型方法构造了一个单连通的光滑闭几乎Ricci平坦非自旋5-流形，并证明了其最小体积为零且在截面曲率有界时发生坍缩。

Almost conformally flat hypersurfaces

We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact $n$-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a consequence, we determine the homology of almost conformally flat hypersurfaces. Furthermore, we provide a necessary condition for a compact Riemannian manifold to admit an isometric minimal immersion as a hypersurface in the sphere and extend a result due to Shiohama and Xu \cite{SX} for compact hypersurfaces in any space form.

具有负曲率的光滑内在平坦极限

A Smooth Intrinsic Flat Limit of with Negative Curvature

针对Gromov关于内在平坦收敛是否保持非负标量曲率的问题，本文构造反例，证明正标量曲率流形的内在平坦极限可具有负标量曲率。

Subspace foliations and collapse of closed flat manifolds

We study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov-Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular.

A quantum Leray-Hirsch theorem for banded gerbes

For a gerbe $\Y$ over a smooth proper Deligne-Mumford stack $\B$ banded by a finite group $G$, we prove a structure result on the Gromov-Witten theory of $\Y$, expressing Gromov-Witten invariants of $\Y$ in terms of Gromov-Witten invariants of $\B$ twisted by various flat $U(1)$-gerbes on $\B$. This is interpreted as a Leray-Hirsch type of result for Gromov-Witten theory of gerbes.

Noncollapsed degeneration of Einstein 4-manifolds I

A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs of Ricci-flat ALE spaces. This raises some natural issues, in particular: can all Einstein orbifolds be Gromov-Hausdorff limits of smooth Einstein manifolds? Can we describe more precisely the smooth Einstein metrics close to a given singular one? In this first paper, we prove th…

Observable concentration of mm-spaces into nonpositively curved manifolds

The measure concentration property of an mm-space $X$ is roughly described as that any 1-Lipschitz map on $X$ to a metric space $Y$ is almost close to a constant map. The target space $Y$ is called the screen. The case of $Y=\mathbb{R}$ is widely studied in many literature (see \cite{gromov}, \cite{ledoux}, \cite{mil2}, \cite{milsch}, \cite{sch}, \cite{tal}, \cite{tal2} and its reference). M. Gromov developed the theory of measure concentration in the case where the screen $Y$ is not necessarily $\mathbb{R}$ (cf. \cite{gromovcat}, {gromov2}, \cite{gromov}). In this paper, we consider the case …

On $L_2$-cohomology of almost Hermitian manifolds

Let $(X,J,ω,g)$ be a complete $n$-dimensional Kähler manifold. A Theorem by Gromov \cite{G} states that the if the Kähler form is $d$-bounded, then the space of harmonic $L_2$ forms of degree $k$ is trivial, unless $k=\frac{n}{2}$. Starting with a contact manifold $(M,α)$ we show that the same conclusion does not hold in the category of almost Kähler manifolds. Let $(X,J,g)$ be a complete almost Hermitian manifold of dimension four. We prove that the reduced $L_2$ $2^{nd}$-cohomology group decomposes as direct sum of the closure of the invariant and anti-invariant $L_2$-cohomology. This genera…

Collapsing sequences of solutions to the Ricci flow on 3-manifolds with almost nonnegative curvature

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilating about infinite time singularities. In this case we classify the possible Gromov-Hausdorff limits and construct 2-dimensional virtual limits. The virtual limits are constructed using Fukaya theory of the limits of local covers. We then show that the virtual limit arising from a…

Chern-Ricci流与t-Gauduchon Ricci平坦条件

Chern-Ricci flow and t-Gauduchon Ricci-flat condition

本文研究Chern-Ricci流下的t-Gauduchon Ricci平坦条件，提供了不保持该条件的例子、非平凡新解及几何流的Gromov-Hausdorff极限行为，明确描述了特定主丛上pluriclosed流的极限。

Gromov-Hausdorff collapsing of Calabi-Yau manifolds

This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the base manifold. Secondly, if the fibered Calabi-Yau manifolds are Lagrangian fibrations of holomorphic symplectic manifolds, the metrics on the regular parts of the limits are special Kähler metrics. By combining these two results, we extend arXiv:math/0008018 to any fibered projecti…

Quasi-Kähler Chern-flat manifolds and complex 2-step nilpotent Lie algebras

The study of quasi-Kähler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras. In the present paper we show that quasi-Kähler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitian structures satisfying the second Gray identity. From an algebraic point of view this correspondence reads as a natural correspondence between anti-bi-invariant almost complex structures on Lie algebras to bi-invariant complex structures. Some natural algebraic problems are approached and…

源自Kähler–Einstein边缘度量的Eguchi–Hanson度量

Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics

本文在Calabi–Hirzebruch流形上构造沿两个除子奇异的Kähler–Einstein边缘度量，研究锥角取极值时的Gromov–Hausdorff极限，证明Eguchi–Hanson度量自然出现，并完整描述所有极限行为，解决了Cheltsov–Rubinstein猜想。

关于局部对偶平坦和射影平坦的几乎有理Finsler度量

On Locally Dually and Projectively Flat Almost Rational Finsler Metrics

研究几乎有理Finsler度量的局部对偶平坦和射影平坦的充要条件，并刻画同时满足两者的度量，提出局部Minkowski性猜想。

不具备实射影或平坦共形结构的流形

Manifolds without real projective or flat conformal structures

该研究通过相对严格双曲化方法构造了维数≥5的闭光滑流形实例，这类流形既无实射影也无平坦共形结构，且基本群为非初等Gromov双曲群，丰富了相关流形的研究案例。

Asymptotically flat Fredholm bundles and assembly

Almost flat finitely generated projective Hilbert C*-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a K-homology class $η\in K_*(M)$ twisted with an almost flat bundle in terms of the image of $η$ under Lafforgue's assembly map and the almost representation associated to the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture. We show how to gener…

通过调和扭曲旋量的几何刚性

Geometric Rigidity via Almost-Harmonic Twisted Spinors

研究Gromov精确提升二形式方法在标量曲率几何中的应用，通过扭曲L^2指标构造调和旋量，证明锐利双曲标量曲率比较，并分析等式情形得到原度量是Einstein的。

Volume growth of 3-manifolds with scalar curvature lower bounds

We give a new proof of a recent result of Munteanu--Wang relating scalar curvature to volume growth on a $3$-manifold with non-negative Ricci curvature. Our proof relies on the theory of $μ$-bubbles introduced by Gromov as well as the almost splitting theorem due to Cheeger--Colding.

Collapsed manifolds with local Ricci bounded covering geometry

For $ρ, v>0$, we say that an $n$-manifold $M$ satisfies local $(ρ,v)$-bound Ricci covering geometry, if Ricci curvature $\text{Ric}_M\ge -(n-1)$, and for all $x\in M$, $\text{vol}(B_ρ(\tilde x))\ge v>0$, where $\tilde x$ is an inverse image of $x$ on the (local) Riemannian universal cover of the $ρ$-ball at $x$. In this paper, we extend the nilpotent fiber bundle theorem of Cheeger-Fukaya-Gromov on a collapsed $n$-manifold $M$ of bounded sectional curvature to $M$ of a local $(ρ,v)$-bound Ricci covering geometry, and $M$ is close to a non-collapsed Riemannian manifold of lower dimension. The n…

Singular positive mass theorem with arbitrary ends

Motivated by the recent progress on positive mass theorem for asymptotically flat manifolds with arbitrary ends and the Gromov's definition of scalar curvature lower bound for continuous metrics, we start a program on the positive mass theorem for asymptotically flat manifolds with $C^0$ arbitrary ends. In this work as the first step, we establish the positive mass theorem of asymptotically flat manifolds with $C^0$ arbitrary ends when the metric is $W^{1,p}_{\mathrm{loc}}$ for some $p\in(n,\infty]$ and is smooth away from a non-compact closed subset with Hausdorff dimension $n-\frac{p}{p-1}$.…