Publications
Here are my research papers. They are available as PDF files and may differ slightly from the published versions.
General Relativity & Hyperbolic PDE
Preprints
- . Cauchy Data for Multiple Collapsing Boson Stars
We construct Cauchy initial data for the Einstein–Maxwell–Klein–Gordon (EMKG) system that evolve in finite time into spacetimes containing multiple trapped surfaces. From a physical perspective, this corresponds to preparing multiple well-separated boson stars, each of which collapses to form a spacelike black hole region. In particular, this work extends earlier results on the formation of multiple trapped surfaces in vacuum to the Einstein–Maxwell–Klein–Gordon system.
- . Mass-Centered GCM Framework in Perturbations of Kerr(–Newman)
The nonlinear stability problem for black hole solutions of the Einstein equations critically depends on choosing an appropriate geometric gauge. In the vacuum setting, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has played a central role in the proof of stability for slowly rotating Kerr spacetime. In the charged setting, electromagnetic–gravitational coupling destroys the exceptional behavior of the $\ell=1$ center-of-mass mode exploited in the vacuum analysis, causing the standard GCM construction to break down. In this work, we develop an alternative GCM framework, which we call mass-centered, designed to overcome this difficulty and aimed at the nonlinear stability of Reissner–Nordström and Kerr–Newman spacetimes. Our approach replaces transport-based control of the center-of-mass quantity with a sphere-wise vanishing condition imposed on a renormalized $\ell=1$ mode, yielding mass-centered GCM hypersurfaces with modified gauge constraints. The resulting elliptic–transport system remains determined once an $\ell=1$ basis is fixed via effective uniformization, and it provides an alternative construction even in the vacuum (uncharged) limit.
- . Einstein–Maxwell Equations on Mass-Centered GCM Hypersurfaces
The resolution of the nonlinear stability problem for black holes as solutions of the Einstein equations relies crucially on imposing appropriate geometric gauge conditions. In the vacuum setting, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has played a central role in the proof of stability for slowly rotating Kerr spacetime. In the charged setting, electromagnetic–gravitational coupling destroys the exceptional structure exploited in the vacuum analysis, requiring a modified framework. In this work, we solve the Einstein–Maxwell equations on a mass-centered spacelike GCM hypersurface, which is equivalent to solving the constraint equations there. We control all geometric quantities of the solution in terms of suitable seed data corresponding to gauge-invariant fields describing coupled gravitational–electromagnetic radiation in perturbations of Reissner–Nordström or Kerr–Newman spacetimes. These fields, first identified by the second author, are expected to satisfy favorable hyperbolic equations. This result provides a first step toward controlling gauge-dependent quantities in the nonlinear stability analysis of the Reissner–Nordström and Kerr–Newman families.
- . Formation of Multiple Black Holes from Cauchy Data
We construct a class of smooth, asymptotically flat Cauchy initial data for the Einstein vacuum equations that contain no trapped surfaces, but whose future development leads in finite time to the formation of multiple causally independent trapped surfaces and hence multiple black holes. The construction may be viewed as a relativistic analogue of $N$-body initial data: the black holes are not present initially and are not glued in \emph{a priori}, but instead arise dynamically from regular vacuum data through gravitational collapse. The construction combines Christodoulou’s short-pulse method with a surgical gluing procedure for the constraint equations, remaining exactly Brill–Lindquist outside localized regions.
- . Formation of Trapped Surfaces for the Einstein–Maxwell–Charged Scalar Field System
We prove a scale-critical trapped surface formation result for the Einstein–Maxwell–charged scalar field (EMCSF) system without imposing any symmetry assumptions. Starting from regular characteristic initial data prescribed at past null infinity, we establish a semi-global existence theorem at the critical scale and show that the focusing of gravitational waves, the concentration of electromagnetic fields, or the condensation of the complex scalar field can each independently lead to the formation of a trapped surface. In addition, we capture a nontrivial charging process occurring along past null infinity, which introduces new analytical difficulties due to the abnormal behavior of the matter fields. Despite these challenges, both the semi-global existence result and the trapped surface formation remain valid.
- . A Canonical Foliation on Null Infinity in Perturbations of Kerr
Kerr stability for small angular momentum has been established in a series of works by Klainerman–Szeftel, Giorgi–Klainerman–Szeftel, and Shen. Building on the existence and uniqueness theory of GCM spheres developed by Klainerman–Szeftel, we construct a canonical foliation of future null infinity for perturbations of Kerr spacetime. For this foliation, the null energy, linear momentum, center of mass, and angular momentum are well-defined and satisfy the expected physical laws of gravitational radiation. The rigidity of the construction eliminates the usual ambiguities present in the physics literature. We further show that, under the assumptions of the Kerr stability framework, the center of mass of the black hole undergoes a large recoil following the perturbation.
Accepted / To appear
- . Boundedness and Decay for the Teukolsky System in Kerr–Newman Spacetime II: The Case |a| << M, |Q| < M in Axial Symmetry
We establish boundedness and polynomial decay results for the Teukolsky system in the exterior spacetime of very slowly rotating and strongly charged sub-extremal Kerr–Newman black holes, with a focus on axially symmetric solutions. A key step in achieving these results is the derivation of a physical-space Morawetz estimate for the associated generalized Regge–Wheeler system, avoiding the use of spherical harmonic decomposition.
Published
- . Physical-Space Estimates for Axisymmetric Waves on Extremal Kerr Spacetime
We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (Morawetz estimates) through an analysis conducted exclusively in physical space. The boundedness of the energy and Morawetz estimates for axisymmetric waves in extremal Kerr were first obtained by Aretakis through frequency-localized currents, which were used to express the trapping degeneracy. Here, we extend to extremal Kerr a method introduced by Stogin in the sub-extremal case, simplifying Aretakis’ derivation of Morawetz estimates through purely classical currents.
Geometric Analysis
Preprints
- . Constructing Entire Minimal Graphs by Evolving Planes
We introduce an evolving-plane ansatz for the explicit construction of entire minimal graphs of dimension $n$ ($n \ge 3$) and codimension $m$ ($m \ge 2$), for any odd integer $n$. Under this ansatz, the minimal surface system reduces to the geodesic equation on the Grassmannian in affine coordinates. From a geometric perspective, this equation governs how the slope of an $(n-1)$-plane evolves as it sweeps out a minimal graph. This framework yields a rich family of explicit entire minimal graphs of odd dimension $n$ and arbitrary codimension $m$. Moreover, for each such entire minimal graph, its conormal bundle gives rise to an entire special Lagrangian graph in $\mathbb{C}^{n+m}$.
- . Sharp Interior Gradient Estimate for Area-Decreasing Graphical Mean Curvature Flow in Arbitrary Codimension
We establish a sharp interior gradient estimate for area-decreasing graphical mean curvature flow in arbitrary codimension. The result generalizes the classical interior gradient estimate for codimension one to higher codimension settings under the natural area-decreasing condition, extending earlier work in the literature.
Accepted / To appear
- . Rigidity of Contracting Maps Using Harmonic Map Heat Flow
Motivated by recent work of Tsai–Tsui–Wang, we study rigidity phenomena for maps between compact manifolds with positive curvature. We show that a distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result extends to a broader class of manifolds with positive curvature under a weaker contracting condition, interpolating between distance non-increasing and area non-increasing. The analysis is based on the harmonic map heat flow.
Published
- . Rigidity of Area Non-Increasing Maps
In this work, we study area non-increasing maps between manifolds with positive curvature. By employing the strong maximum principle along the graphical mean curvature flow, we show that such maps are either homotopically trivial, Riemannian submersions, local isometries, or isometric immersions, confirming a conjecture of Tsai–Tsui–Wang. In addition, we combine Brendle’s sphere theorem with the mean curvature flow coupled to the Ricci flow to establish related rigidity results on manifolds with positive (1)-isotropic curvature.
- . Preserving Positive Intermediate Curvature
Consider a compact manifold (N) (with or without boundary) of dimension (n). Positive (m)-intermediate curvature interpolates between positive Ricci curvature ((m=1)) and positive scalar curvature ((m=n-1)), and it is obstructed on partial tori (N^n = M^{n-m} \times \mathbb{T}^m). Given Riemannian metrics (g, \tilde g) on ((N, \partial N)) with positive (m)-intermediate curvature and (m)-positive difference (h_g - h_{\tilde g}) of second fundamental forms, we show that there exists a smooth family of Riemannian metrics with positive (m)-intermediate curvature interpolating between (g) and (\tilde g). Moreover, we apply this result to prove a non-existence theorem for partial torical bands with positive (m)-intermediate curvature and strictly (m)-convex boundaries.
- . Uniqueness Theorems of Self-Conformal Solutions to Inverse Curvature Flows
It is known from the literature that round spheres are the only closed homothetic self-similar solutions to the inverse mean curvature flow and parabolic curvature flows defined by degree $-1$ homogeneous functions of principal curvatures in Euclidean space. In this article, we prove a stronger rigidity result: under natural conditions such as star-shapedness, round spheres are the only closed solutions to the aforementioned flows that evolve by diffeomorphisms generated by conformal Killing fields.