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Pengyu Le

Publications and source records attributed to Pengyu Le.

13 recordsLinked to original sources
arXiv2026

Volume-Distance-Ratio Asymptote and Spacetime Inextendibility

This paper develops geometric criteria for determining the inextendibility of spacetimes near singularities based on asymptotic analysis of volume-distance relationships. We introduce and analyze the asymptotic behavior of the volume-distance-ratio (VDR), defined as the ratio of volumes of small chronological diamonds to appropriate powers of distances between their vertices. In $\mathrm{C}^0$ and $\mathrm{C}^{0,1}$ spacetimes (which are weaker than the classical $\mathrm{C}^2$ regularity), we prove that VDR converges to the Minkowski value as chronological diamonds approach accumulation points. The central contribution is the establishment of inextendibility criteria showing that failure of VDR convergence to the Minkowski value implies inextendibility of the spacetime. These criteria apply to spacetime extensions satisfying $\mathrm{C}^0$ locally null-non-accumulating strongly-causal conditions and $\mathrm{C}^{0,1}$ strongly-causal conditions, where the local null-non-accumulation condition is introduced as a fundamental structural property ensuring the validity of VDR-based inextendibility criteria. Concrete applications demonstrate the power and scope of these methods. We prove that $2$-dimensional Misner spacetime is $\mathrm{C}^0$ strongly-causal inextendible and that spatially flat FLRW spacetimes with linear scale factor behavior are $\mathrm{C}^0$ locally null-non-accumulating strongly-causal inextendible. Furthermore, we establish $\mathrm{C}^{0,1}$ strongly-causal inextendibility for Christodoulou's class of spherically symmetric self-similar naked singularity spacetimes.

gr-qc↗
arXiv2026

Volume-Distance-Ratio Asymptote and Spacetime Inextendibility for FLRW Spacetimes

This paper examines the volume-distance-ratio (VDR) asymptote at the past timelike boundary for Friedman-Lemaître-Robertson-Walker (FLRW) spacetimes. We consider spatially flat FLRW spacetimes with scale factor $a(t) \sim t^α$, as well as spatially hyperbolic and spherical FLRW spacetimes with scale factor $a(t) \sim a_0 t^α$. Using criteria for spacetime inextendibility based on the VDR asymptote, we investigate the conditions under which these FLRW spacetimes are past inextendible.

gr-qc↗
arXiv2024

On effective uniformization of 2-sphere and the stability

We provide a proof of effective uniformization for nearly round 2-spheres, utilizing an identity related to the third-order differential of the conformal factor. This identity is connected to the geometry of the embedded spacelike surface within the Minkowski lightcone. Additionally, we investigate the stability of the effective uniformization introduced by Klainerman and Szeftel. Our proof is based on a geometric insight: an isometric embedding of a round sphere into Euclidean space can be constructed using an orthogonal basis of the first eigenspace of the Laplacian operator, with the rectangular coordinates corresponding to the basis functions. By adopting these approaches, we simplify both the proofs of effective uniformization and its stability, while also refining the assumptions underlying both results.

math.DG↗
arXiv2024

Lorentz polarisation and isoperimetric inequality in Minkowski spacetime

In this paper, we prove an isoperimetric inequality for the domain of dependence of a finite lightcone in the Minkowski spacetime of dimension greater than or equal to 3. The inequality involves two quantities: the volume of the domain of dependence, and the perimeter of the finite lightcone. It states that among all finite lightcones with the same perimeter, the maximal volume of the domain of dependence is achieved by the spacelike hyperplane truncated finite lightcone. A novelty of this isoperimetric inequality is the codimension 2 comparison feature. We introduce the Lorentz polarisation to prove the isoperimetric inequality by studying the corresponding variational problem. A key observation is the monotonicity of the domain of dependence of a finite lightcone under the Lorentz polarisation. We show that any finite lightcone can be transformed by Lorentz polarisations to approximate a spacelike hyperplane truncated finite lightcone with an equal or less perimeter. As further applications of the method of Lorentz polarisation, we prove the following isoperimetric type inequalities: a) For a set with the given perimeter in the hyperboloid in the Minkowski spacetime, the geodesic ball in the hyperboloid has the maximal volume of the domain of dependence of the set; b) For an achronal hypersurface with boundary in the lightcone (or the hyperboloid), given the perimeter of the boundary fixed, the spacelike hyperplane disk has the maximal area.

math.DG↗
arXiv2024

Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime

The constant mass function foliation has been shown useful for studying the null Penrose inequality on a null hypersurface, because of the monotonicity formula of Hawking mass along such a foliation. In this paper, we show the global existence of the constant mass aspect function foliation on a nearly spherically symmetric incoming null hypersurface, emanating from a spacelike surface near the apparent horizon to the past null infinity in a vacuum perturbed Schwarzschild spacetime. Moreover, we study the geometry of the constant mass aspect function foliation, by comparing with the spherically symmetric foliation in the Schwarzschild spacetime. The knowledge about the geometry of the foliation is essential for investigating the perturbation of the constant mass aspect function foliation, which is the core in the application to the null Penrose inequality for a vacuum perturbed Schwarzschild spacetime.

math.DG↗
arXiv2023

A Lorentz invariant sharp Sobolev inequality on the circle

We prove the following sharp Sobolev inequality on the circle $$\int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} θ\geq - \frac{4π^2}{\int_{\mathbb{S}^1} v^{-2} \mathrm{d} θ},$$ with the equality being achieved when $v^{-2} (θ) = \frac{k\sqrt{1-α^2}}{1+ α\cos(θ- θ_o)}$where$k>0$, $α\in (-1,1)$, $θ_0 \in \mathbb{R}$. If $v$ vanishes somewhere on the circle, then $$4 \int_{\mathbb{S}^1} (v')^2 \mathrm{d} θ\geq\int_{\mathbb{S}^1} v^2 \mathrm{d} θ.$$ The basic tools to prove the inequality are the rearrangement inequality on the circle and the variational method. We investigate the variational problem of the functional $\mathcal{F}[v] = \int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} θ$ under the constraint $\int_{\mathbb{S}^1} v^{-2} \mathrm{d} θ= 2π$. An important geometric insight of the functional $\mathcal{F}$ is that it is invariant under the Lorentz group, since $\mathcal{F}[v]$ is the integral of the product of two null expansions of a spacelike curve parameterised by the function $v^{-2}$ in a lightcone in $3$-dim Minkowski spacetime. The global minimiser of $\mathcal{F}$ under the constraint is simply given by the spacelike plane section of the lightcone. We introduce a method which combines the symmetric decreasing rearrangement and the Lorentz transformation. This method isnot confined to the scope of this paper, but is applicable to other Lorentz invariant variational problems on $\mathbb{S}^{n}, n \geq 1$. As an example, we sketch a proof of the sharp Sobolev inequality on $\mathbb{S}^n, n\geq 3$ by this method.

math.FA↗
arXiv2023

Linearised perturbation of constant mass aspect function foliation in Schwarzschild black hole spacetime

We study the linearised perturbation of the constant mass aspect function foliation in a Schwarzschild black hole spacetime. In particular, we investigate the linearised perturbation of the asymptotic geometry of the foliation at null infinity. We show that there is a 4-dimensional linear space for the linearised perturbation of the initial leaf inside the event horizon, corresponding to which the linearised perturbation of the asymptotic geometry of the foliation at null infinity is preserved to be round. For such a linearised perturbation of the initial leaf in this 4-dimensional linear space, we calculate the corresponding linearised perturbation of the energy-momentum vector and the Bondi mass at null infinity. We show that the linearised perturbations of the Bondi energy and the Bondi mass both vanish, and every possible linearised perturbation of the linear momentum can be achieved by a linearised perturbation of the initial leaf in the 4-dimensional linear space.

math.DG↗
arXiv2022

Marginally trapped surfaces in a perturbed Schwarzschild spacetime

The concept of a marginally trapped surface is important in the theory of general relativity. In the Schwarzschild black hole spacetime, its event horizon is foliated by marginally trapped surfaces. In a more general black hole spacetime, the concept of a marginally trapped surface is closely related to various sorts of horizon, for example, the apparent horizon, the trapping boundary, the isolated horizon and the dynamical horizon. In this paper, we study the set of marginally trapped surfaces in a perturbed Schwarzschild spacetime. We show that for every incoming null hypersurface which is nearly spherically symmetric, there exists a unique embedded marginally trapped surface. In order to prove this result, we develop a general method to study the geometry of spacelike surfaces in a double null coordinate system, which can be applied to study other problems for spacelike surfaces in a Lorentzian manifold.

math.DG↗
arXiv2022

Global regular null hypersurfaces in a perturbed Schwarzschild black hole exterior

The spherically symmetric null hypersurfaces in a Schwarzschild spacetime are smooth away from the singularities and foliate the spacetime. We prove the existence of more general foliations by null hypersurfaces without the spherical symmetry condition. In fact we also relax the spherical symmetry of the ambient spacetime and prove a more general result: in a perturbed Schwarzschild spacetime (not necessary being vacuum), nearly round null hypersurfaces can be extended regularly to the past null infinity, thus there exist many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole. A significant point of the result is that the ambient spacetime metric is not required to be differentiable in all directions.

math.DG↗
arXiv2022

Marginal tubes and foliations by marginal surfaces

In this paper, we introduce the notion of a marginal tube, which is a hypersurface foliated by marginal surfaces. It generalises the notion of a marginally trapped tube and several notions of black hole horizons, for example trapping horizons, isolated horizons, dynamical horizons, etc. We prove that if every spacelike section of a marginal tube is a marginal surface, then the marginal tube is null. There is no assumption on the topology of the marginal tube. To prove it, we study the geometry of spacelike surfaces in a 4-dimensional spacetime with the help of double null coordinate systems. The result is valid for arbitrary 4-dimensional spacetimes, regardless of any energy condition.

math.DG↗
arXiv2021

Note on strong cosmic censorship for spherically symmetric Einstein-dust model

In this note, we derive an ordinary differential equation for outgoing light rays in the spacetime of a spherically symmetric contracting dust cloud. The violation of strong cosmic censorship is equivalent to the existence of a solution blowing up at the centre of the symmetry. As an application, we reprove the generic violation of strong cosmic censorship for a dust cloud first proved by Christodoulou. We also derive a similar equation for a spherically symmetric dust cloud in the case of positive cosmological constant and show the generic violation of strong cosmic censorship in this case.

gr-qc↗
arXiv2017

The intersection of a hyperplane with a lightcone in the Minkowski spacetime

Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For the intrinsic geometry of the intersection, depending on the hyperplane being spacelike, null or timelike, it has the constant positive, zero or negative Gaussian curvature. For the extrinsic geometry of the intersection, we find that it is a noncompact marginal trapped surface when the hyperplane is null. In this case, we find a geometric interpretation of the Green's function of the Laplacian on the standard sphere. In the end, we contribute a clearer understanding of the anisotropic criterion for formation of trapped surfaces in vacuum.

math.DG↗