arXiv subjects

Hyun Chul Jang

Publications and source records attributed to Hyun Chul Jang.

10 recordsLinked to original sources

arXiv2026

Propagation of Chaos on Riemannian Manifolds

We extend reflection coupling for uniform-in-time propagation of chaos from Euclidean space to complete Riemannian manifolds. Intrinsic reflection along minimizing geodesics replaces the Euclidean difference process, and the associated endpoint-index term quantifies the competition between curvature, confinement, and interaction. Under an endpoint-index bound and a radial Laplacian bound, we establish global well-posedness, uniform moment estimates, and uniform-in-time propagation of chaos with rate $N^{-1/2}$ in a modified Wasserstein distance. We further obtain exponential contraction of the nonlinear McKean--Vlasov semigroup, uniqueness of its invariant probability measure among probability measures with finite second moment, and exponential convergence to equilibrium. Although a Ricci lower bound is an important special case, our framework also includes a curvature-spike class with $\inf_M\operatorname{Ric}=-\infty$.

math.PR↗

arXiv2026

A spacetime positive mass theorem with corners via mollification

We prove a strict dominant energy deformation theorem for asymptotically flat initial data with corners along a hypersurface $Σ$. The deformation preserves a corner condition on the Bartnik data across $Σ$. We show that if the dominant energy condition holds on each side of $Σ$ and the Bartnik data satisfy this corner condition, then the exterior end satisfies $E \ge |P|$ in every dimension $n \ge 3$.

math.DG↗

arXiv2026

Entropy Stability for products of negatively curved symmetric spaces

Let $(M,g_0)$ be a closed oriented $n$-manifold that is locally isometric to a product $(X^{n_1}_1,g_1)\times\cdots (X_k^{n_k},g_k)$, where each $n_i\ge 3,$ and each factor $(X_i^{n_i},g_i)$ is a negatively curved symmetric space. We study the stability of minimal entropy rigidity for such manifolds. Specifically, we consider whether an entropy-minimizing sequence $(M,g_i)$ converges to the model space in the measured Gromov-Hausdorff sense after removing negligible subsets. Previously, Song [Son23] established this type of stability for negatively curved symmetric spaces, where both the $n$-volume of the removed subsets and the $(n-1)$-volume of their boundaries converge to zero. We construct a counterexample demonstrating that this stronger stability notion does not generally hold in the product case; in particular, the condition that the $(n-1)$-volume of the boundary of removed subsets converges to zero cannot be imposed. Nonetheless, we prove that an entropy-minimizing sequence $(M,g_i)$ converges to the model space after removing subsets whose $n$-volume converges to zero in the measured Gromov-Hausdorff topology. This result provides a weaker form of stability compared to the negatively symmetric case. A key ingredient in establishing this stability is our proof of the intrinsic uniqueness of the spherical Plateau solution for products of negatively curved symmetric spaces, which is of independent interest.

math.DG↗

arXiv2025

Deterministic--Distance Couplings of Brownian Motions on Radially Isoparametric Manifolds

We develop a unified geometric framework for coadapted Brownian couplings on radially isoparametric manifolds (RIM)--spaces whose geodesic spheres have principal curvatures $κ_1(r),\dots,κ_{n-1}(r)$ depending only on the geodesic radius $r$. The mean curvature of such a geodesic sphere is denoted by $A(r) = \mathrm{Tr}(S_r) = \sum_{i=1}^{n-1} κ_i(r)$, where $S_r$ is the shape operator of the sphere of radius $r$. Within the stochastic two--point Itô formalism, we derive an intrinsic drift--window inequality \[ A(r) - \sum_i |κ_i(r)| \;\le\; ρ'(t) \;\le\; A(r) + \sum_i |κ_i(r)|, \] governing the deterministic evolution of the inter--particle distance $ρ_t = d(X_t, Y_t)$ under all coadapted couplings. We prove that this bound is both necessary and sufficient for the existence of a coupling realizing any prescribed distance law $ρ(t)$, thereby extending the constant--curvature classification of Pascu--Popescu (2018) to all RIM. The endpoints of the drift window correspond to the synchronous and reflection couplings, providing geometric realizations of extremal stochastic drifts. Applications include stationary fixed--distance couplings on compact--type manifolds, linear escape laws on asymptotically hyperbolic spaces, and rigidity of rank--one symmetric geometries saturating the endpoint bounds. This establishes a direct correspondence between radial curvature data and stochastic coupling dynamics, linking Riccati comparison geometry with probabilistic coupling theory.

math.PR↗

arXiv2025

Rigidity of Asymptotically Hyperboloidal Initial Data Sets with Vanishing Mass

In Special Relativity, massless objects are characterized as either vacuum states or as radiation propagating at the speed of light. This distinction extends to General Relativity for asymptotically flat initial data sets (IDS) $(M^n, g, k)$, where vacuum is represented by slices of Minkowski space, and radiation is modeled by slices of $pp$-wave spacetimes. In contrast, we demonstrate that asymptotically hyperboloidal IDS with zero mass must embed isometrically into Minkowski space, with no possible IDS configurations modeling radiation in this setting. Our result holds under the most general assumptions. The proof relies on precise decay estimates for spinors on level sets of spacetime harmonic functions and works in all dimensions.

math.DG↗

arXiv2022

Scalar curvature deformation and mass rigidity for ALH manifolds with boundary

We study scalar curvature deformation for asymptotically locally hyperbolic (ALH) manifolds with nonempty compact boundary. We show that the scalar curvature map is locally surjective among either (1) the space of metrics that coincide exponentially toward the boundary, or (2) the space of metrics with arbitrarily prescribed nearby Bartnik boundary data. Using those results, we characterize the ALH manifolds that minimize the Wang-Chruściel-Herzlich mass integrals in great generality and establish the rigidity of the positive mass theorems.

math.DG↗

arXiv2022

Hyperbolic mass via horospheres

We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has scalar curvature lower bound -n(n-1) and is isometric to hyperbolic space outside a coordinate horosphere, then the manifold is isometric to hyperbolic space. In addition, we apply our formula to investigate regions near infinity that do not contribute to the mass quantity, which leads to improved rigidity results of hyperbolic space.

math.DG↗

arXiv2019

Mass rigidity for hyperbolic manifolds

We prove the rigidity of positive mass theorem for asymptotically hyperbolic manifolds. Namely, if the mass equality holds, then the manifold is isometric to hyperbolic space. The result was previously proven for spin manifolds or under special asymptotics.

math.DG↗

arXiv2019

Some scalar curvature warped product splitting theorems

We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on marginally outer trapped surfaces applied to appropriate initial data sets. One of the results involves an analysis of Obata's equation on manifolds with boundary. This result is relevant to recent work of Lan-Hsuan Huang and the second author concerning the rigidity of asymptotically locally hyperbolic manifolds with zero mass.

math.DG↗

arXiv2018

Asymptotically Hyperbolic 3-Metric with Ricci flow foliation

In general relativity, there have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. In particular, C. -Y. Lin used a foliation by the Ricci flow on 2-spheres to establish an asymptotically flat extension and C. Sormani and Lin proved useful results with this extension. In this paper, we construct asymptotically hyperbolic 3-metrics with the Ricci flow foliation. We also study the rigid case when the Hawking mass of the inner surface of the manifold agrees with its total mass.

math.DG↗