arXiv subjects

Puskar Mondal

Publications and source records attributed to Puskar Mondal.

At least 19 recordsLinked to original sources

arXiv2026

Compact Proof of the Positivity of Quasi-Local Masses for a class of Initial Data

We prove a purely quasi-local positivity theorem for the Wang--Yau mass for a class of initial data whose Jang deformation, after a boundary-preserving conformal reduction to zero scalar curvature, lies in a sufficiently small transverse--traceless (TT) perturbative neighborhood of a strictly convex Euclidean fill-in.We explicitly construct a nontrivial class of physical initial data whose admissible Jang reductions realize this TT-generated sector. The argument reduces the Wang--Yau energy to the Brown--York mass of the resulting scalar-flat compact metric, together with nonnegative bulk terms determined by the Jang deformation, and establishes strict positivity by computing the second variation of the Brown--York functional at the Euclidean metric in transverse--traceless directions. The proof is entirely confined to the compact fill-in and uses neither an asymptotically flat extension nor the positive mass theorem. This gives a partial answer to a question of R. Schoen concerning a genuinely quasi-local proof of positivity for quasi-local mass.

math.DG↗

arXiv2026

Boundary-Geometry-Driven Black Hole Formation in Vacuum

We identify a boundary-geometry-driven mechanism for the dynamical formation of marginally outer trapped surfaces (MOTSs) in vacuum general relativity. Mild three-dimensional anisotropies evolve inside a compact Cauchy domain whose effective isotropic thickness remains controlled, while the generalized boundary mean curvature can increase during either contracting or expanding boundary evolution. This drives the boundary across Yau's geometric threshold, forcing MOTS formation from initially untrapped data. We further interpret the characteristic shear construction of Ref.~\cite{MondalYau2026} as the null manifestation of the same anisotropic vacuum dynamics. The result provides a purely vacuum physical realization of MOTS formation through global geometric effects, without invoking a short-pulse concentration mechanism for gravitational radiation.

gr-qc↗

arXiv2026

Formation of trapped surfaces for the spherically symmetric Einstein-Yang-Mills system with non-trivial incoming data

We establish a trapped surface formation theorem for the spherically symmetric Einstein Yang Mills equations in a double-null gauge. The theorem concerns characteristic initial data posed on a pair of transversely intersecting null hypersurfaces and allows nontrivial incoming data. The proof extends the singular characteristic method of An and Lim for the Einstein Maxwell Charged Scalar Field System to the non-abelian Yang Mills setting, where the curvature coupling and gauge field nonlinearities introduce new structural difficulties. This paper constitutes the first part of a program toward weak cosmic censorship for the Einstein Yang Mills system.

gr-qc↗

arXiv2026

Volume Stability for Hyperbolic Manifolds and Applications to General Relativity

We prove a sharp volume-stability theorem for closed hyperbolic three-manifolds. Let $(M,h)$ be closed hyperbolic with $\operatorname{Ric}_h=-2h$, and let $g_i$ be smooth metrics on $M$ satisfying $R(g_i)\geq -6, \operatorname{Vol}_{g_i}(M)\longrightarrow \operatorname{Vol}_h(M)$. After passing to a subsequence, there exist $Z_i\subset M$, smooth domains $K_i\subset M$, and diffeomorphisms $ψ_i:K_i\longrightarrow M\setminus Z_i $ such that $\operatorname{Vol}_{g_i}(Z_i)\longrightarrow0, \operatorname{Vol}_h(M\setminus K_i)\longrightarrow0, $ and $ \|ψ_i^*g_i-h\|_{C^0(K_i,h)}\longrightarrow0. $ Thus near-equality in the sharp hyperbolic volume bound forces tensorial $C^0$-convergence to the hyperbolic metric outside regions of vanishing volume. As an application, we establish stability of the Fischer--Moncrief reduced Hamiltonian at the Lorentz-cone ground state: after CMC normalization, near-minimizing compact vacuum data in the hyperbolic topological class converge, modulo sets of vanishing volume, to the hyperbolic Lorentz-cone geometry in tensorial $C^0$. This provides a rigorous volume-dominance formulation of the Fischer--Moncrief asymptotic picture.

math.DG↗

arXiv2026

Semi-Global Existence and Trapped Surface Formation for the Einstein-Vlasov System

In this article, we study the Einstein-Vlasov system for massless particles. Our main contribution is the provision of a novel method for controlling the Vlasov matter in a double null gauge, relying purely on vector field commutation and bypassing the need for introducing Jacobi fields, which has so far been the only existing technique outside of symmetry in this gauge. To obtain it, we have to overcome stringent regularity issues that exist along this path. Because it relies solely on vector field commutation, our technique is fundamentally simple and flexible enough to be applicable in any data regime. We thus anticipate this to be a helpful tool for many subsequent problems regarding the Einstein-Vlasov system, including, for instance, simplified approaches to the proof of nonlinear stability of Minkowski spacetime with massless Vlasov matter, scattering problems for data close to black holes among others. Within the present double-null commutator framework, we show that two derivatives of curvature suffice to close the coupled Einstein--Vlasov hierarchy, without derivative loss, given smooth initial data. Using this method, we obtain a large data semi-global existence theorem and a dynamical trapped surface formation statement for our system.

math.AP↗

arXiv2026

Dynamical Formation of Black Holes due to Boundary Effect in Vacuum Gravity

We prove the dynamical formation of a marginally outer trapped surface in pure vacuum spacetime from smooth asymptotically flat Cauchy data which initially contain no MOTS. The mechanism is a boundary effect rather than a collapse mechanism. We work in a Cauchy--double-null framework and use Yau's boundary criterion \cite{yau}, which gives the existence of an interior MOTS from a lower bound for the generalized boundary mean curvature relative to the Schoen--Yau radius of the domain. We construct an explicit class of vacuum initial data for which this criterion is strictly subcritical on the initial hypersurface, while the Einstein evolution drives the same domain into the supercritical regime. More precisely, a mild incoming gravitational radiation field increases the generalized boundary mean curvature of an isotropically large interior region sufficiently to force the formation of a MOTS in its future development. A characteristic feature of the initial data is a large interior anisotropic curvature component: the trace-free Ricci curvature is of larger order than the scalar curvature, which is balanced at the vacuum constraint scale. Thus the MOTS forms not from matter concentration or standard gravitational collapse, but from the interaction between boundary geometry, large-scale interior geometry, and the vacuum Einstein dynamics. This gives a rigorous realization of a long-suspected physical idea that apparent horizons may form from global geometric effects in vacuum general relativity.

gr-qc↗

arXiv2026

A large data result for vacuum Einstein's equations

We prove a global well-posedness and asymptotic convergence theorem for the $(3+1)$-dimensional vacuum Einstein equations with positive cosmological constant $Λ$ on globally hyperbolic spacetimes $\widetilde M \cong M \times \mathbb R$, where $M$ is a closed three-manifold of negative Yamabe type. In constant-mean-curvature transported spatial coordinates, an open set of large initial data gives rise to future-global solutions whose renormalized spatial metrics converge smoothly to a limiting metric of constant negative scalar curvature. The key new ingredient is an integrable damping mechanism, induced by the cosmological constant in this gauge and absent in the $Λ=0$ vacuum problem, which yields time-integrable decay for the nonlinear evolution. As a consequence, the Einstein--$Λ$ flow does not in general canonically encode the Thurston geometrization of the underlying three-manifold. This confirms a conjecture of Ringström on the asymptotic topological indistinguishability of large-data Einstein--$Λ$ dynamics. An analogous theorem is also proved for manifolds of positive Yamabe type, under an additional technical hypothesis.

gr-qc↗

arXiv2025

On a Rigidity Result in Positive Scalar Curvature Geometry

I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that geodesic balls of radii strictly less than $\fracπ{2}$ in $n+1~(n\geq 2)$ dimensional unit sphere can be rigid under smooth deformations that increase scalar curvature preserving the intrinsic geometry and the mean curvature of the boundary, and such rigidity result fails for the hemisphere. The proof of this assertion requires the notion of a real Killing connection and solution of the boundary value problem associated with its Dirac operator. The result serves as the sharpest refinement of the now-disproven Min-Oo conjecture.

math.DG↗

arXiv2024

Quasi-local masses in General relativity and their positivity: Spinor approach

We study the quasi-local masses arising in general relativity using spinors and prove their positivity property. This leads to the question of a pure quasi-local proof of the positivity of the Wang-Yau \cite{yau} quasi-local mass. More precisely we prove that the gravitational mass bounded by a spacelike topological $2-$sphere is non-negative in a generic spacetime verifying dominant energy condition and vanishes only if the surface is embedded in the Minkowski space. This construction is purely quasi-local in nature and in particular does not rely on Bartanik's gluing and asymptotic extension construction \cite{bartnik1993quasi} and subsequent application of the positive mass theorem \cite{schoen1979proof,schoen1981proof} to prove the positivity of quasi-local mass. The result involves solving Dirac equation on a compact Riemannian manifold with boudary using MIT Bag and APS boundary condition.

math-ph↗

arXiv2024

A new conformal quasi-local energy in general relativity

We construct new conserved quasi-local energies in general relativity using the formalism developed by \cite{CWY}. In particular, we use the optimal isometric embedding defined in \cite{yau,yau1} to transplant the conformal Killing fields of the Minkowski space back to the $ 2-$ surface of interest in the physical spacetime. For an asymptotically flat spacetime of order $1$, we show that these energies are always finite. Their limit as the total energies of an isolated system is evaluated and a conservation law under Einsteinian evolution is deduced.

gr-qc↗

arXiv2024

Global exterior stability of the Minkowski space: coupled Einstein-Yang-Mills perturbations

Here we prove a global gauge-invariant radiation estimates for the perturbations of the $3+1$ dimensional Minkowski spacetime in the presence of Yang-Mills sources. In particular, we obtain a novel gauge invariant estimate for the Yang-Mills fields coupled to gravity in a double null framework in the Causal complement of a compact set of a Cauchy slice. A consequence of our result is the global exterior stability of the Minkowski space under coupled Yang-Mills perturbations. A special structure present both in the null Bianchi equations and the null Yang-Mills equations is utilized crucially to obtain the dispersive estimates necessary to conclude the global existence property. Direct use of Bel-Robinson and Yang-Mills stress-energy tensor to obtain the energy estimates is avoided in favor of weighted integration by parts taking advantage of the manifestly symmetric hyperbolic characteristics of null Bianchi and null Yang-Mills equations. Our result holds for any compact semi-simple gauge group. This is the first stability result of Minkowski space including a non-linear source.

gr-qc↗

arXiv2024

Big-bang limit of $2+1$ gravity and Thurston boundary of Teichmüller space

We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type $Σ_{p}\times \mathbb{R}$, $p>1$, where $Σ_{p}$ is a closed Riemann surface of genus $p$, in the regime of $2+1$ dimensional classical general relativity. The configuration space of the gauge fixed dynamics is identified with the Teichmüller space ($\mathcal{T}Σ_{p}\approx \mathbb{R}^{6p-6}$) of $Σ_{p}$. Utilizing the properties of the Dirichlet energy of certain harmonic maps, estimates derived from the associated elliptic equations in conjunction with a few standard results of the theory of the compact Riemann surfaces, we prove that every non-trivial solution curve runs off the edge of the Teichmüller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations ($\mathcal{PML}$ $\mathcal{PMF}$), the Thurston boundary of the Teichmüller space.

gr-qc↗

arXiv2023

A Geometric Approach to the Yang-Mills Mass Gap

I provide a new idea based on geometric analysis to obtain a positive mass gap in pure non-abelian renormalizable Yang-Mills theory. The orbit space, that is the space of connections of Yang-Mills theory modulo gauge transformations, is equipped with a Riemannian metric that naturally arises from the kinetic part of reduced classical action and admits a positive definite sectional curvature. The corresponding regularized \textit{Bakry-Émery} Ricci curvature (if positive) is shown to produce a mass gap for $2+1$ and $3+1$ dimensional Yang-Mills theory assuming the existence of a quantized Yang-Mills theory on $(\mathbb{R}^{1+2},η)$ and $(\mathbb{R}^{1+3},η)$, respectively. My result on the gap calculation, described at least as a heuristic one, applies to non-abelian Yang-Mills theory with any compact semi-simple Lie group in the aforementioned dimensions. In $2+1$ dimensions, the square of the Yang-Mils coupling constant $g^{2}_{YM}$ has the dimension of mass, and therefore the spectral gap of the Hamiltonian is essentially proportional to $g^{2}_{YM}$ with proportionality constant being purely numerical as expected. Due to the dimensional restriction on $3+1$ dimensional Yang-Mills theory, it seems one ought to introduce a length scale to obtain an energy scale. It turns out that a certain `trace' operation on the infinite-dimensional geometry naturally introduces a length scale that has to be fixed by measuring the energy of the lowest glu-ball state. However, this remains to be understood in a rigorous way.

hep-th↗

arXiv2023

Continuation Criterion For Solutions To The Einstein Equations

We prove a continuation condition in the context of 3+1 dimensional vacuum Einstein gravity in Constant Mean extrinsic Curvature (CMC) gauge. More precisely, we obtain quantitative criteria under which the physical spacetime can be extended in the future indefinitely as a solution to the Cauchy problem of the Einstein equations given regular initial data. In particular, we show that a gauge-invariant $H^{2}$ Sobolev norm of the spacetime Riemann curvature remains bounded in the future time direction provided the so-called deformation tensor of the unit timelike vector field normal to the chosen CMC hypersurfaces verifies a spacetime $L^\infty$ bound. To this end, we implement a novel technique to obtain this refined estimate by using Friedlander's parametrix for tensor wave equations on curved spacetime and Moncrief's subsequent improvement. We conclude by providing a physical explanation of our result as well as its relation to the issues of determinism and weak cosmic censorship.

gr-qc↗

arXiv2023

Einstein-Yang-Mills equations in the double null framework

We prove a semi-global gauge-invariant estimate for the solutions of the characteristic initial value problem associated with the coupled Einstein-Yang-Mills equations. In particular, we prove the existence of \textit{a} future development of regular initial data on a pair of incoming and outgoing null hypersurfaces emanating from a spacelike topological $2$-sphere. This marks the first study of the characteristic initial value problem of Einstein's equations with a non-linear source.

gr-qc↗

arXiv2023

Mass and infinite dimensional geometry

I unravel an elegant geometric meaning of the mass of the lowest energy excited state of a renormalizable quantized field theory by studying the weighted geometry of the classical configuration space of the theory. A suitably defined regularized Bakry-Emery Ricci curvature of these infinite dimensional spaces controls the spectra of the corresponding quantum Hamiltonians. The Ricci curvature part of the full Bakry-Emery Ricci curvature appears to be purely quantum in nature. This geometric contribution to the spectra in the context of quantum field theory has not been studied previously to my knowledge. Assuming the existence of rigorous quantization, I present a few problems starting from massive free particles to the non-abelian Yang-Mills theory. A remarkable property is observed in the large $N$ Yang-Mills theory, where a non-trivial mass gap is preserved. This occurs due to the fact that the regularized Bakry-Emery Ricci curvature that is responsible for the gap of the configuration space scales as $g^{2}_{YM}N=λ$ ('t Hooft coupling) that remains invariant.

hep-th↗

arXiv2023

Formation of trapped surfaces in the Einstein-Yang-Mills system

We prove a scale-invariant, semi-global existence result and a trapped surface formation result in the context of coupled Einstein-Yang-Mills theory, without symmetry assumptions. More precisely, we prove a scale-invariant semi-global existence theorem from past null-infinity and show that the focusing of the gravitational and/or chromoelectric-chromomagnetic waves could lead to the formation of a trapped surface. Adopting the signature for decay rates approach introduced in \cite{A19}, we develop a novel gauge (and scale) invariant hierarchy of non-linear estimates for the Yang-Mills curvature which, together with the estimates for the gravitational degrees of freedom, yields the desired semi-global existence result. Once semi-global existence has been established, the formation of a trapped surface follows from a standard ODE argument.

math.AP↗

arXiv2022

On the Global Well-Posedness of the Einstein-Yang-Mills System

In this paper, we present a partial result on the global well-posedness of the Cauchy problem for the Einstein-Yang-Mills system in the constant mean extrinsic curvature spatial harmonic and generalized Coulomb gauges as introduced in [Mondal, arXiv:2112.14273]. We give a small-data global existence theorem for a family of $n+1$ dimensional spacetimes with $n\geq4$, utilizing energy arguments presented in [Andersson and Moncrief, arXiv:0908.0784]. We observe that these energy arguments will fail for $n=3$ due to the conformal invariance of the $3+1$ Yang-Mills equations and present a gauge-covaraiant formulation of the Einstein-Yang-Mills system in $3+1$ dimensions to show that an energy argument cannot be used to prove the global well-posedness result, regardless of the choice of gauge.

gr-qc↗