Searcharxiv⌕ Search

arXiv subjects

Conghan Dong

Publications and source records attributed to Conghan Dong.

10 recordsLinked to original sources
arXiv2026

The Penrose conjecture for initial data sets satisfying a $2$-convexity condition

Let $(M^3, g, \mathbf{k})$ be a smooth, connected, asymptotically flat initial data set with connected outermost past apparent horizon $\Sigma $. We prove the Penrose conjecture, namely that $m_{\mathrm{ADM}}(g) \geq \sqrt{\frac{|\Sigma |}{16 \pi }} $, under the assumptions of the dominant energy condition and the $2$-convexity condition that the sum of the two smallest eigenvalues of $\mathbf{k}$ is nonnegative. The main tool is the $\sigma $-inverse mean curvature flow, together with a monotonicity formula developed in \cite{Dong26SigmaIMCF}.

math.DG↗
arXiv2026

On $3$-manifolds with small mass and $L^2$-curvature

One of S.T. Yau's problems asks the following: given a $3$-dimensional asymptotically flat manifold $M$ with non-negative scalar curvature and $L^2$-norm of the curvature tensor at most $1$, if the mass of $M$ is small, is there a bilipschitz diffeomorphism from $M$ to the flat Euclidean space $\mathbb{R}^3$? We provide a strong positive answer to this problem by using our previous work \cite{DS25}.

math.DG↗
arXiv2026

The $\sigma$-inverse mean curvature flow and the generalized Penrose conjecture

Let $(M^3, g, \mathbf{k})$ be a complete asymptotically flat initial data set satisfying the dominant energy condition, and let $m$ denote its ADM mass. The generalized Penrose conjecture asserts that the area of an outermost generalized apparent horizon $N\subset M$ satisfies $|N| \leq 16 \pi m^2$. In this paper, we establish this inequality for each connected component of $N$ in the case where $\mathbf{k}$ is proportional to the metric $g$. Our approach is based on a new geometric evolution, which we call the $\sigma $-inverse mean curvature flow, together with a novel monotonicity formula that may be of independent interest.

math.DG↗
arXiv2024

Non-collapsing of Ricci shrinkers with bounded curvature

We establish a uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and a uniform curvature bound. Additionally, we extend the non-collapsing result to a broader class of smooth metric measure spaces satisfying Bakry-\'Emery conditions.

math.DG↗
arXiv2024

Some stability results of positive mass theorem for uniformly asymptotically flat $3$-manifolds

In this paper, we show that for a sequence of orientable complete uniformly asymptotically flat $3$-manifolds $(M_i , g_i)$ with nonnegative scalar curvature and ADM mass $m(g_i)$ tending to zero, by subtracting some open subsets $Z_i$, whose boundary area satisfies $\mathrm{Area}(\partial Z_i) \leq Cm(g_i)^{1/2 - \varepsilon}$, for any base point $p_i \in M_i\setminus Z_i$, $(M_i\setminus Z_i,g_i,p_i)$ converges to the Euclidean space $(\mathbb{R}^3,g_E,0)$ in the $C^0$ modulo negligible volume sense. Moreover, if we assume that the Ricci curvature is uniformly bounded from below, then $(M_i, g_i, p_i)$ converges to $(\mathbb{R}^3,g_E,0)$ in the pointed Gromov-Hausdorff topology.

math.DG↗
arXiv2024

Stability for the 3D Riemannian Penrose inequality

We show that the Schwarzschild 3-manifold is stable for the 3-dimensional Riemannian Penrose inequality in the pointed measured Gromov-Hausdorff topology, modulo negligible domains and boundary area perturbations.

math.DG↗
arXiv2023

Stability of Euclidean 3-space for the positive mass theorem

We show that the Euclidean 3-space $\mathbb{R}^3$ is stable for the Positive Mass Theorem in the following sense. Let $(M_i,g_i)$ be a sequence of complete asymptotically flat $3$-manifolds with nonnegative scalar curvature and suppose that the ADM mass $m(g_i)$ of one end of $M_i$ converges to $0$. Then for all $i$, there is a subset $Z_i$ in $M_i$ such that $M_i\setminus Z_i$ contains the given end, the area of the boundary $\partial Z_i$ converges to zero, and $(M_i\setminus Z_i,g_i)$ converges to $\mathbb{R}^3$ in the pointed measured Gromov-Hausdorff topology for any choice of basepoints. This confirms a conjecture of G. Huisken and T. Ilmanen. Additionally, we find an almost quadratic upper bound for the area of $\partial Z_i$ in terms of $m(g_i)$. As an application of the main result, we also prove R. Bartnik's strict positivity conjecture.

math.DG↗
arXiv2021

$W^{1,p}$-metrics and conformal metrics with $L^{n/2}$-bounded scalar curvature

A $W^{1,p}$-metric on an $n$-dimensional closed Riemannian manifold naturally induces a distance function, provided $p$ is sufficiently close to $n$. If a sequence of metrics $g_k$ converges in $W^{1,p}$ to a limit metric $g$, then the corresponding distance functions $d_{g_k}$ subconverge to a limit distance function $d$, which satisfies $d\le d_g$. As an application, we show that the above convergence result applies to a sequence of conformal metrics with $L^{n/2}$-bounded scalar curvatures, under certain geometric assumptions. In particular, in this special setting, the limit distance function $d$ actually coincides with $d_{g}$.

math.DG↗
arXiv2018

Convergence of manifolds under some $L^p$-integral curvature conditions

Let $\mathcal{C}(\mathcal{R},n,p,Λ,D,V_0)$ be the class of compact $n$-dimensional Riemannian manifolds with finite diameter $\leq D$, non-collapsing volume $\geq V_0$ and $L^p$-bounded $\mathcal{R}$-curvature condition $\|\mathcal{R}\|_{L^p}\leq Λ$ for some $p>\frac n2$. Let $(M,g_0)$ be a compact Riemannian manifold and $\mathcal{C}(M,g_0)$ the class of manifolds $(M,g)$ conformal to $(M,g_0)$. In this paper we use $\varepsilon$-regularity to show a rigidity result in the conformal class $\mathcal{C}(S^n,g_0)$ of standard sphere under $L^p$-scalar rigidity condition. Then we use harmonic coordinate to show $C^α$-compactness of the class $\mathcal{C}(K,n,p,Λ,D,V_0)$ with additional positive Yamabe constant condition, where $K$ is the sectional curvature, and this result will imply a generalization of Mumford's lemma. Combining these methods together we give a geometric proof of $C^α$-compactness of the class $\mathcal{C}(K,n,p,Λ,D,V_0)\cap \mathcal{C}(M,g_0)$. By using Weyl tensor and a blow down argument, we can replace the sectional curvature condition by Ricci curvature and get our main result that the class $\mathcal{C}(Ric,n,p,Λ,D,V_0)\cap \mathcal{C}(M,g_0)$ has $C^α$-compactness.

math.DG↗