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Pei-Ken Hung

Publications and source records attributed to Pei-Ken Hung.

17 recordsLinked to original sources
arXiv2024

Thom's gradient conjecture for nonlinear evolution equations

R. Thom's gradient conjecture states that if a gradient flow of an analytic function converges to a limit, it does so along a unique limiting direction. In this paper, we extend and settle this conjecture in the context of infinite dimensional problems. Building on the foundational works of Łojasiewicz, L. Simon, and the resolution of the conjecture for finite dimensional cases by Kurdyka-Mostowski-Parusinski, we focus on nonlinear evolutions on Riemannian manifolds as studied by L. Simon. This framework includes geometric PDEs such as minimal surface, harmonic map, mean curvature flow, and normalized Yamabe flow. Our main result not only confirms the uniqueness of the limiting direction but also characterizes the rate of convergence and possible limiting directions for both classical and infinite dimensional settings.

math.AP↗
arXiv2024

Boundary Behavior of Compact Manifolds With Scalar Curvature Lower Bounds and Static Quasi-Local Mass of Tori

A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature, must have total mean curvature not greater than that of the isometric embedding into Euclidean 3-space, with equality only for domains in this reference manifold. We generalize this result to 2-tori of Guass curvature greater than $-1$, which bound a compact 3-manifold having scalar curvature not less than $-6$ and at least one other boundary component satisfying a 'trapping condition'. The conclusion is that the total weighted mean curvature is not greater than that of an isometric embedding into the Kottler manifold, with equality only for domains in this space. Examples are given to show that the assumption of a secondary boundary component cannot be removed. The result gives a positive mass theorem for the static Brown-York mass of tori, in analogy to the Shi-Tam positivity of the standard Brown-York mass, and represents the first such quasi-local mass positivity result for non-spherical surfaces. Furthermore, we prove a Penrose-type inequality in this setting.

math.DG↗
arXiv2023

Asymptotics for slowly converging evolution equations

We investigate slowly converging solutions for non-linear evolution equations of elliptic or parabolic type. These equations arise from the study of isolated singularities in geometric variational problems. Slowly converging solutions have previously been constructed assuming the Adams-Simon positivity condition. In this study, we identify a necessary condition for slowly converging solutions to exist, which we refer to as the Adams-Simon non-negativity condition. Additionally, we characterize the rate and direction of convergence for these solutions. Our result partially confirms Thom's gradient conjecture in the context of infinite-dimensional problems.

math.AP↗
arXiv2022

The Positive Energy Theorem for Asymptotically Hyperboloidal Initial Data Sets With Toroidal Infinity and Related Rigidity Results

We establish the positive energy theorem and a Penrose-type inequality for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity, weakly trapped boundary, and satisfying the dominant energy condition. In the umbilic case, a rigidity statement is proven showing that the total energy vanishes precisely when the initial data manifold is isometric to a portion of the canonical slice of the associated Kottler spacetime. Furthermore, we provide a new proof of the recent rigidity theorems of Eichmair-Galloway-Mendes [10] in dimension 3, with weakened hypotheses in certain cases. These results are obtained through an analysis of the level sets of spacetime harmonic functions.

math.DG↗
arXiv2021

A Minkowski-type inequality in the AdS-Melvin space

The AdS-Melvin spacetime was introduced by Astorino and models the AdS soliton with electromagnetic charge. It is a static spacetime with a time-symmetric Cauchy hypersurface, which we refer to as the AdS-Melvin space. In this paper, we study a sharp Minkowski-type inequality for surfaces embedded in the AdS-Melvin space. We first prove the inequality for special cases in which the surface enjoys axisymmetry or is a small perturbation of a coordinate torus. We then use a weighted normal flow to show that the inequality holds for general surfaces.

math.DG↗
arXiv2021

Determinantal structures for Bessel fields

A Bessel field $\mathcal{B}=\{\mathcal{B}(α,t), α\in\mathbb{N}_0, t\in\mathbb{R}\}$ is a two-variable random field such that for every $(α,t)$, $\mathcal{B}(α,t)$ has the law of a Bessel point process with index $α$. The Bessel fields arise as hard edge scaling limits of the Laguerre field, a natural extension of the classical Laguerre unitary ensemble. It is recently proved in [LW21] that for fixed $α$, $\{\mathcal{B}(α,t), t\in\mathbb{R}\}$ is a squared Bessel Gibbsian line ensemble. In this paper, we discover rich integrable structures for the Bessel fields: along a time-like or a space-like path, $\mathcal{B}$ is a determinantal point process with an explicit correlation kernel; for fixed $t$, $\{\mathcal{B}(α,t),α\in\mathbb{N}_0\}$ is an exponential Gibbsian line ensemble.

math.PR↗
arXiv2021

A Minkowski inequality for Horowitz-Myers geon

We prove a sharp inequality for toroidal hypersurfaces in three and four dimensional Horowitz-Myers geon. This extend previous results on Minkowski inequality in the static spacetime to toroidal surfaces in asymptotically hyperbolic manifold with flat toroidal conformal infinity.

math.DG↗
arXiv2019

The linear stability of the Schwarzschild spacetime in the harmonic gauge: even part

In this paper we study the even part of the linear stability of the Schwarzschild spacetime as a continuation of [22]. By taking the harmonic gauge, we prove that the energy decays at a rate $τ^{-2+}$ for the solution of the linearized Einstein equation after subtracting its spherically symmetric part. We further show that the spherically symmetric part converges to a linear combination of two special solutions. One is the gauge-fixed mass change solution [19]. The other is the deformation tensor of a stationary one form, which solves the tensorial wave equation. As a key ingredient, we prove that the solutions of the tensorial wave equation converge to this stationary one form up to a scalar multiplication.

gr-qc↗
arXiv2019

Inverse Mean Curvature Flow with Singularities

This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each singular point. We prove the cone also evolves by the inverse mean curvature flow and each singularity is removed when the evolving cone becomes flat. As a result, we derive the exact waiting time for a weak solution to be a smooth solution. In particular, a necessary and sufficient condition for an existence of smooth classical solution is given.

math.DG↗
arXiv2018

Linear Stability of Schwarzschild Spacetime: Decay of Metric Coefficients

In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region. We employ Hodge decomposition to split the solution into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For the co-closed portion, we extend previous results by the first two authors, deriving Regge-Wheeler type equations for two gauge-invariant master quantities without the earlier paper's need of axisymmetry. For the closed portion, we build upon earlier work of Zerilli and Moncrief, wherein the authors derive an equation for a gauge-invariant master quantity in a spherical harmonic decomposition. We work with gauge-invariant quantities at the level of perturbed connection coefficients, with the initial value problem formulated on Cauchy data sets. With the choice of an appropriate gauge in each of the two portions, decay estimates on these decoupled quantities are used to establish decay of the metric coefficients of the solution, completing the proof of linear stability. Our result differs from that of Dafermos-Holzegel-Rodnianski, both in our choice of gauge and in our identification and utilization of lower-level gauge-invariant master quantities.

gr-qc↗
arXiv2018

Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities

In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild met- rics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively cor- respond to the closed and co-closed, or polar and axial, solutions in the case of four spacetime dimensions, while the two-tensor portion is a new feature in the higher-dimensional setting. Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge decomposition, we produce decoupled gauge-invariant master quantities satisfying Regge- Wheeler type wave equations in each of the three portions. The scalar and co-vector quantities respectively generalize the Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four spacetime dimen- sions; beyond these quantities, we further discover a higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the co-vector por- tion. In the analysis of the master quantities, we strengthen the mode stability result of Kodama-Ishibashi to a uniform boundedness estimate in all dimensions; further, we prove decay estimates in the case of six or fewer spacetime dimensions. Finally, we provide a rigorous argument that linearized solutions of low angular frequency are decomposable as a sum of pure gauge solution and linearized Myers-Perry solution, the lat- ter solutions generalizing the linearized Kerr solutions in four spacetime dimensions.

gr-qc↗
arXiv2018

The linear stability of the Schwarzschild spacetime in the harmonic gauge: odd part

In this paper, we study the odd solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of Regge-Wheeler quantities, we are able to estimate the odd part of Lichnerowicz d'Alembertian equation. In particular, we prove the solution decays at rate $τ^{-1+δ}$ to a linearlized Kerr solution.

math.DG↗
arXiv2016

Linear stability of Schwarzschild spacetime subject to axial perturbations

In this paper, we address the issue of linear stability of Schwarzschild space- time subject to certain axisymmetric perturbations. In particular, we prove that associ- ated solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, decay to a linearized Kerr metric. Our method employs a complex line bundle interpretation applied to a connection-level object, allow- ing for direct analysis of this connection-level object by the linearized Einstein equations, in contrast with the recent breakthrough of Dafermos-Holzegel-Rodnianski.

math.DG↗
arXiv2016

The rest mass of an asymptotically Anti-de Sitter spacetime

We study the space of Killing fields on the four dimensional AdS spacetime $AdS^{3,1}$. Two subsets $\mathcal{S}$ and $\mathcal{O}$ are identified: $\mathcal{S}$ (the spinor Killing fields) is constructed from imaginary Killing spinors, and $\mathcal{O}$ (the observer Killing fields) consists of all hypersurface orthogonal, future timelike unit Killing fields. When the cosmology constant vanishes, or in the Minkowski spacetime case, these two subsets have the same convex hull in the space of Killing fields. In presence of the cosmology constant, the convex hull of $ \mathcal{O}$ is properly contained in that of $\mathcal{S}$. This leads to two different notions of energy for an asymptotically AdS spacetime, the spinor energy and the observer energy. In [10], Chruściel, Maerten and Tod proved the positivity of the spinor energy and derived important consequences among the related conserved quantities. We show that the positivity of the observer energy follows from the positivity of the spinor energy. A new notion called the "rest mass" of an asymptotically AdS spacetime is then defined by minimizing the observer energy, and is shown to be evaluated in terms of the adjoint representation of the Lie algebra of Killing fields. It is proved that the rest mass has the desirable rigidity property that characterizes the AdS spacetime.

math.DG↗
arXiv2014

A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold

We prove a sharp inequality for hypersurfaces in the n-dimensional Anti-deSitter-Schwarzschild manifold for general n greater or equal to 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by the first author in [3].

math.DG↗