Neckpinch — Physics

A complete parameter classification of left-invariant Bott affine Ricci solitons for two rotated distributions on three-dimensional Lorentzian Lie groups

Mon, 26 Oct 2026 00:00:00 GMT

Fanqi Lin — We classify all left-invariant affine Ricci solitons associated with the Bott connections of two fixed rank-two distributions on the seven standard families of three-dimensional Lorentzian Lie groups. The distributions are defined by a fixed orthogonal change of the spacelike frame. A universal adapted-frame calculation provides explicit connection matrices, curvature operators and the symmetrized Ricci tensor. The equation in the orthogonal-complement direction forces every left-invariant solution to be steady. The remaining equations yield elementary parameter families and finitely many real algebraic branches: four normalized branches for G₂, six branches for each sign parameter in G₄, and two quartic branches in G₇ in addition to its elementary families. Resultants, subresultants, exact polynomial reduction and Sturm root isolation establish necessity, sufficiency and exhaustiveness. Degenerate cases are treated separately, including all families with nonunique soliton vector fields. An invertible frame transformation relates the two distribution problems. The classification is in the stated structure parameters; no quotient by isometries, automorphisms or scaling is taken.

Ricci flow of Integrable Sigma Models, Hitchin Systems and Quantum Seiberg-Witten theory

Thu, 22 Oct 2026 00:00:00 GMT

G. Bonelli, S. Lacroix, T. Pedroni, A. Tanzini — We construct a family of integrable 2d sigma models through 4d Chern-Simons theory and show that their classical point-particle reduction corresponds to a Hitchin system. The integrable Hitchin quantisation then provides a natural candidate Hamiltonian for the one-loop reduced dynamics of the sigma models, which coincides with the Perelman operator in all cases examined. This allows us to study the corresponding spectral problems using 4d 𝓝=2 gauge theory and 2d conformal field theory methods. For the sausage model, we derive exact quantisation conditions and compute exponentially small corrections essential for the agreement with numerical finite-volume results.

Using Ricci Flow in the Construction of Homogeneous Cosmological Models

Mon, 19 Oct 2026 00:00:00 GMT

Fan Zhang, Lee Lindblom — This study develops numerical methods for computing homogeneous and isotropic cosmological models on manifolds with non-trivial spatial topologies. The use of Ricci flow is introduced here to aid in the construction of globally homogeneous initial data on these manifolds. The initial data produced in this way are evolved by solving Einstein's equation numerically to construct homogeneous and isotropic cosmological models on three exemplar manifolds in this study: Seifert-Weber space, the lens space L(5,2), and Sixth-Turn space.

Algebraic and Symmetrized Affine star-Ricci Solitons for the Bott Connection on Three-Dimensional Lorentzian Lie Groups

Wed, 14 Oct 2026 00:00:00 GMT

Yuxin Ding — We fix a spacelike two-dimensional left-invariant distribution and its associated almost contact structure on a three-dimensional Lorentzian Lie group, and investigate the star-Ricci contraction of the curvature of the Bott connection. We compute the complete tensor for seven standard bracket families and determine all real parameter solutions of the general algebraic equation and its self-adjoint subclass. Using the symmetric part of the tensor, we also define a Bott affine star-Ricci soliton equation and classify all left-invariant vector field solutions. Every affine solution is steady, while two non-unimodular families admit branches with a nonzero symmetrized tensor. The proofs provide complete component equations, treat all degenerate parameter branches, and establish sufficiency by explicit substitution. We also distinguish almost contact structures from contact metric structures.

A Positive Mass Type Theorem for a Singular Toroidal Static Model

Wed, 23 Sep 2026 00:00:00 GMT

Zhixin Wang — We study positive mass and related Brown–York type inequalities for three-dimensional manifolds modeled on a static space with flat toroidal slices. Using inverse mean curvature flow, we first derive an inequality relating the asymptotic geometry at infinity to the size of an interior singularity. This inequality admits a natural interpretation as a positive mass type theorem. We then combine this global inequality with a Shi–Tam type construction to obtain a Brown–York type inequality for boundaries isometric to flat tori. In contrast to the Schwarzschild setting, our argument does not extend directly to general boundary surfaces, and we identify several obstructions to such a generalization. Finally, we discuss examples illustrating the relationship between negative mass, the strength of interior singularities, and the topology of the interior.

A continuous Positive Mass Theorem for perturbations of Euclidean space

Tue, 22 Sep 2026 00:00:00 GMT

Paula Burkhardt-Guim — We prove a Positive Mass Theorem for C⁰-asymptotically flat Riemannian metrics with nonnegative scalar curvature in a weak sense that are sufficiently uniformly close to Euclidean space. More precisely, we show that a C⁰-asymptotically flat Riemannian metric that is a C⁰ perturbation of Euclidean space with nonnegative scalar curvature in the sense of Ricci flow has nonnegative mass, where the mass is given by a C⁰ analog of the classical ADM mass previously introduced by the author.

Weighted normalized curve shortening flow with applications in pseudo-Euclidean spaces

Mon, 21 Sep 2026 00:00:00 GMT

Baichuan Hu, Xiang Ma — The focus of this paper is the curve shortening flow for closed spacelike curves in pseudo-Euclidean spaces, which has very few results so far. Will they produce singularities where certain tangent line tends to light cone? If not, will such a curve shrink to a circular point? To answer these questions, we establish a dichotomy for planar weighted normalized curve shortening flow with uniformly positive and bounded weights. Applying to closed smooth spacelike curves in pseudo-Euclidean spaces that admit a one-to-one convex projection onto a spacelike plane, at their finite maximal time we will see: either the curve shrinks to a point and becomes asymptotically circular, or the tangent directions subsequentially approach the null cone. Both alternatives occur. In the first case, this proves our previous conjecture that a strong spacelike curve in ℝ^(2,q) with index 1 will converge to a circular point under the usual CSF. In the latter case, a monotone area-bivector defect is found in ℝ^(2,1), which gives a quantitative obstruction to point collapse. Explicit examples of spacelike curves with lightlike tangent limit are given.

Conformal Killing–Yano Ricci solitons: Structure, compatibility, and rigidity

Sun, 20 Sep 2026 00:00:00 GMT

Mohammadjavad Habibivostakolaei, Abbas M. Sherif, Yen-Kheng Lim — We introduce a geometric structure – a conformal Killing–Yano Ricci soliton (CKY–RS) – that couples conformal Ricci soliton (CRS) geometry to conformal Killing–Yano (CKY) 2–forms. The soliton field of the CRS geometry is given by the divergence of the CKY 2–form. We introduce a conserved CKY–Cotton current and derive a compatibility identity relating the Cotton tensor, the CRS obstruction tensor, and the CKY 2–form. In 4–dimensional Lorentzian signature, we show that, under non-degeneracy and closedness assumptions on the CKY form, a CKY–RS structure forces the conformal representative to be locally Kerr–NUT–(A)dS. For a closed non-degenerate CKY on a Kerr–NUT–(A)dS background, the conformal deformation is necessarily trivial. For Einstein backgrounds of arbitrary dimension and signature, the conformal factor satisfies an eigenvalue equation and an Obata–type Hessian equation. If the background is also compact or a CKY orbit is periodic, the conformal factor is an invariant of the CKY–flow and we obtain simple spectral obstructions to non-trivial CKY–RS structures. From the Hessian equation, we obtain obstruction and classification results for the non-trivial conformal sector, including product/Brinkmann geometries and a Weyl–aligned branch. Finally, we give explicit constructions for static spherically symmetric geometries and BTZ backgrounds, including a CKY–RS realization with a time-dependent conformally flat representative. These results provide a geometric framework for studying CRS with hidden symmetry structure, with potential applications to exact geometries in general relativity.

Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow

Wed, 16 Sep 2026 00:00:00 GMT

Yuyue Chen, Shiying Wu — We establish the eventual monotonicity of the Willmore deficit ∫_ΣH² dμ−16π along the mean curvature flow of smooth closed embedded strictly convex surfaces in ℝ³, thereby complementing Huisken's exponential convergence with an eventual monotonicity property. As a consequence, the Hawking mass is eventually nondecreasing along the contracting mean curvature flow, in contrast with the usual inclusion monotonicity expected of the Hawking mass. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.

Lorentzian Algebraic Ricci Solitons on the Heisenberg Group

Tue, 15 Sep 2026 00:00:00 GMT

Youssef Ayad — There exist three nonequivalent left invariant Lorentzian metrics on the Heisenberg group H₂ₙ₊₁, or equivalently, three nonequivalent Lorentzian inner products on the Heisenberg Lie algebra h₂ₙ₊₁, denoted by μ, ν, and φ. We show that, in a specific case, μ is an algebraic Ricci soliton that is shrinking. Moreover, ν is an algebraic Ricci soliton only on the three-dimensional Heisenberg Lie algebra h₃ and it is shrinking. Finally, we show that φ is a steady algebraic Ricci soliton on h_(2n + 1) for n > 1. However, for n = 1, φ is flat.

Soliton Solutions to the Curvature Flow on the 2-dimensional De Sitter Space and Applications

Thu, 27 Aug 2026 00:00:00 GMT

Fábio Nunes da Silva, Edwin Salinas Reyes — We show that the spacelike solutions to the curvature flow for curves on the De Sitter space are in correspondence with the solutions to the inverse curvature flow on the 2-dimensional hyperbolic space, that on the De Sitter space, the timelike solutions to the curvature flow are in correspondence with the timelike solutions to the inverse curvature flow, and that the solutions curve shortening flow on the 2-dimensional hyperbolic space are in correspondence with the solutions to the spacelike solutions to the inverse curvature flow on the De Sitter space. We prove that, for spacelike curves on the De Sitter space, the curvature flow is a gradient-type flow for the arc-length functional. We observe that a spacelike or timelike curve on the De Sitter space is a soliton solution to the curvature flow (resp. inverse curvature flow) if and only if its curvature (resp. inverse of its curvature) can be written as the inner product between its tangent vector field and a fixed vector v of the 3-dimensional Minkowski space. We prove that for each vector v, there exists a 2-parameter family of timelike (spacelike) soliton solutions to the curvature flow and to the inverse curvature flow on the De Sitter space. We show that there exists no non-trivial complete timelike soliton. There exist non-trivial complete spacelike solutions. As a consequence of curvature flow, we obtain the behavior of the soliton solutions to the inverse curvature flow on the De Sitter space and hyperbolic space.

Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity

Thu, 20 Aug 2026 00:00:00 GMT

Dake Li, Zhizhang Wang, Shiqi Yin — This paper studies self-shrinkers and the long-time behavior of the inverse σₖ curvature flow for noncompact entire spacelike strictly convex hypersurfaces in Minkowski space. In contrast to the co-compact setting, where the corresponding self-shrinker is rigid, the noncompact problem admits a rich family of self-shrinkers determined by their asymptotic data. More precisely, we formulate the self-shrinker equation as a fully nonlinear Dirichlet problem on hyperbolic space with prescribed data on its ideal boundary, and prove that every continuous negative boundary value determines a unique entire spacelike strictly convex self-shrinker. We further study the inverse σₖ curvature flow starting from an entire spacelike strictly convex hypersurface satisfying a subsolution condition at infinity and a uniform positive lower bound for its σₖ curvature. We prove global existence of the flow and show that the normalized flow converges locally smoothly to the unique self-shrinker with the same prescribed boundary value at infinity.

Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

Tue, 11 Aug 2026 00:00:00 GMT

Tianyin Ren, Chong Song — In this paper, we observe that the (spacelike) mean curvature flow of a submanifold in a (pseudo-)Euclidean space is equivalent to a harmonic-Ricci flow with coupling constant α=−1 (or +1), for the corresponding Gauss map and the induced metric. The solitons of these two flows are also equivalent. As an application, we get a monotonicity formula for the spacelike mean curvature flow.

Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups

Sat, 08 Aug 2026 00:00:00 GMT

Youssef Ayad — In this paper, we compute the canonical and Kobayashi-Nomizu connections, together with their curvature, on Lorentzian four-dimensional nilpotent Lie groups endowed with a product structure. We also classify the algebraic Ricci solitons associated with these connections.

Conformal-Mapping Method for Horizon Multipoles in Numerical Relativity: Implementation, Kerr Validation, and Applications Beyond Axisymmetry

Sat, 08 Aug 2026 00:00:00 GMT

Yeong-Bok Bae, Young-Hwan Hyun, Gungwon Kang — We present a numerical method that constructs geometrically defined coordinates on black-hole horizons, from which the multipole moments are computed without assuming axisymmetry. This method, which we denote the conformal-mapping method (CMM), provides a numerical realization of the conformal construction proposed by Ashtekar et al. in 2022, combining discrete Ricci flow, spectral embedding onto the unit sphere, and Möbius gauge fixing by the vanishing-area-dipole condition. We first test the CMM against analytic Kerr benchmarks, and then apply it to an equal-mass, non-spinning binary black-hole merger. We also compare it with an approximate-symmetry-based method. The CMM allows the multipole moments to be expressed in a fixed reference frame, whereas the symmetry-adapted frame can reorient abruptly when the preferred approximate axis changes. In a frame aligned with the orbital angular momentum, the amplitude of the quadrupole mode grows during inspiral and decays after merger, displaying a qualitative ringdown behavior. These results show that the CMM is a useful tool for studying horizon geometry in dynamical situations where no stable symmetry axis is available.

The Penrose inequality with charge for 2-convex initial data sets

Fri, 31 Jul 2026 00:00:00 GMT

Tuan Dolmen — We prove the Penrose inequality with charge under the 2-convexity condition recently introduced by Dong. More precisely, given a complete, connected and asymptotically flat Einstein-Maxwell initial data set (M,g,k; E,B) satisfying the charged dominant energy and the 2-convexity conditions, with divergence-free electromagnetic vector fields (E,B) and a connected outermost past apparent horizon Σ that satisfies |Σ| ≥ 4πq² - where q is the total charge - we show that the following inequality for the ADM mass m holds: m≥ √((|Σ|)/16π) + q² √(fracπ|Σ|), with equality if and only if k ≡ 0 and (M,g;E,B) is isometric to a canonical slice of sub-extremal Reissner-Nordström spacetime. Building on Dong's proof of the uncharged case, we use his P-inverse mean curvature flow and its weak formulation, which only depends on (g,P) and hence applies to the charged setting unchanged. The novelty of our work is the modification of the monotonicity formula to account for the additional charge term. For time-symmetric data (k ≡ 0), the flow reduces to the classical inverse mean curvature flow and our monotonicity formula to Jang's monotonicity of the charged Hawking mass, recovering the charged Riemannian Penrose inequality.

The Penrose conjecture for initial data sets satisfying a 2-convexity condition

Wed, 22 Jul 2026 00:00:00 GMT

Conghan Dong — Let (M³, g, k) be a smooth, connected, asymptotically flat initial data set with connected outermost past apparent horizon Σ. We prove the Penrose conjecture, namely that m_ADM(g) ≥ √(|Σ|)/(16 π), under the assumptions of the dominant energy condition and the 2-convexity condition that the sum of the two smallest eigenvalues of k is nonnegative. The main tool is the σ-inverse mean curvature flow, together with a monotonicity formula developed in.

Spacelike Brakke flows with boundary in pseudo-Euclidean space

Sat, 11 Jul 2026 00:00:00 GMT

Javier Hervás — We develop a weak formulation of spacelike mean curvature flow in pseudo-Euclidean space. The framework is based on spacelike integer rectifiable varifolds and a pseudo-Euclidean version of first variation. Particular attention is paid to the presence of a fixed spacelike boundary. We introduce a generalized mean curvature vector and a weak spacelike conormal along the boundary. Under natural uniform spacelikeness and curvature assumptions, we prove a closure theorem for spacelike varifolds with boundary. We then define spacelike Brakke flows by means of the corresponding pseudo-Euclidean Brakke inequality, and prove a compactness theorem for sequences of spacelike Brakke flows with fixed boundary. Basic monotonicity properties are established, reflecting the sign structure of the ambient indefinite metric. Finally, we adapt Ilmanen's elliptic regularization procedure to prove existence of spacelike Brakke flows, and the local regularity theorem of White to the pseudo-Euclidean setting. The results provide a varifold setting for studying weak spacelike mean curvature flow beyond singularities.

Entropy and Non-Collapse in Lorentzian Geometry

Fri, 10 Jul 2026 00:00:00 GMT

Rohit Dhormare — In this paper, we establish a geometric correspondence between the Lorentzian Raychaudhuri equation and Perelman's non-collapsing theorem for the Ricci flow. By interpreting the Raychaudhuri equation as a Lorentzian analogue of Ricci flow, we connect geodesic focusing in general relativity to the monotonicity and entropy functionals of geometric analysis. Using this correspondence, we derive a Lorentzian non-collapsing theorem and introduce a covariant entropy functional governing causal volume evolution. Finally, we propose the concept of geodesic entropy capacity, a curvature-bounded limit on the information that can be stored in spacetime regions, providing a unified geometric framework linking gravitation, thermodynamics, and information.

Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow

Tue, 07 Jul 2026 00:00:00 GMT

Jihui Wang, Matteo Lulli, Antonino Marciano — We investigate the stochastic Ricci flow of spherically symmetric perturbations of the Schwarzschild–Anti de Sitter black-hole metric. Elaborating on the Ricci-flow analysis of Headrick and Wiseman, we include a negative cosmological constant through a Ricci-target term and study how the flow is correlated with the thermodynamic heat capacity of the black hole. Numerical simulations show that, in the positive heat-capacity regime, perturbations of the angular sector of the metric relax toward the Schwarzschild–Anti de Sitter fixed point, while in the negative heat-capacity regime they grow under the deterministic Ricci flow. We then introduce a multiplicative stochastic noise and find that sufficiently strong stochasticity can suppress the growth of these perturbations, effectively stabilizing configurations that would otherwise be thermodynamically unstable. Finally, we reformulate the dynamics in terms of an entropy variable evolving on a thermodynamic free-energy landscape, and support the metric-flow results through Monte Carlo simulations and the associated Fokker–Planck equation. These results suggest that stochastic fluctuations can modify the relation between geometric stability under Ricci flow and thermodynamic stability in asymptotically Anti de Sitter black-hole spacetimes.

On the Stability of Einstein Manifolds with Boundary

Mon, 06 Jul 2026 00:00:00 GMT

Rasmus Jouttijärvi — We study the stability problem for Einstein manifolds with boundary with respect to the Einstein-Hilbert action. The geometric boundary conditions we are using arise naturally from studying the calculus of variations associated with Ricci flow. Upon the introduction of a boundary, the space of TTg tensors no longer arise naturally as the defining space for the stability condition. Thus we must settle for the larger subspace of tensors preserving the scalar curvature, the total volume and the Bianchi gauge condition, which we call the space of TVg tensors. As a test-case, we shall discuss stability of the Riemannian Schwarzschild anti-deSitter family of metrics; a problem known as "a Black hole in a box".

Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets

Thu, 02 Jul 2026 00:00:00 GMT

Jacopo Tenan — We construct an exhaustive family of constant spacetime mean curvature (STCMC) surfaces for initial data sets close to the anti-de Sitter-Schwarzschild hyperboloid. In particular, we obtain such a foliation as the long time limit of the volume preserving spacetime mean curvature flow starting from the constant mean curvature foliation constructed by Neves-Tian (Geom. Funct. Anal., 2009). As an application, inspired by the definition of STCMC center of mass for initial data sets proposed in the asymptotically Euclidean setting by Cederbaum-Sakovich (Calc. Var. PDE, 2021), we study the center of mass of an asymptotically hyperboloidal initial data set.

Bochner Flatness and Soliton Dynamics in Lorentzian Kähler Spacetime Geometry

Wed, 24 Jun 2026 00:00:00 GMT

Karthika Ramasamya, Lavanya Kumara, Soumendu Roy, Bülent Ünal — We study Riemann solitons and η-hyperbolic Ricci solitons on Bochner-flat Lorentzian Kähler spacetime manifolds. Under the Einstein field equations with cosmological constant and perfect fluid assumptions, explicit formulas for the soliton parameter are derived, yielding criteria for shrinking, steady, and expanding behaviors. Several physically relevant models, including dark fluid, stiff matter, dust, and radiation, are analyzed. We show that Bochner-flat Lorentzian Kähler spacetimes are Einstein and investigate the resulting geometric and dynamical consequences. In the context of generalized Robertson–Walker spacetimes, we obtain constraints on the warping function and classify soliton solutions. Global properties such as geodesic completeness, singularity formation, and stability are also examined.

Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

Mon, 22 Jun 2026 00:00:00 GMT

Man-Chun Lee, Florian Litzinger, Miles Simon — In this work, we study the positive mass theorem under critical low regularity assumptions using Ricci flow smoothing. We show that asymptotically flat manifolds (Mⁿ,g) of regularity L^∞∩ W^(1,n) with non-negative distributional scalar curvature have non-negative ADM mass. Furthermore, when the ADM mass vanishes, the manifold is globally isometric to Euclidean space with respect to an integral distance introduced by De Cecco-Palmieri. This extends the recent work of Hafemann to the critical regularity case. Our approach is based on showing that Riemannian metrics of regularity L^∞∩ W^(1,n), whose scalar curvature is bounded from below in the distributional sense, admit a Ricci flow smoothing whose scalar curvature is bounded from below by the same initial lower bound in the classical sense. In contrast, Cecchini-Frenck-Zeidler constructed examples of metrics which are in L^∞∩ W^(1,p) for all 2

The σ-inverse mean curvature flow and the generalized Penrose conjecture

Tue, 26 May 2026 00:00:00 GMT

Conghan Dong — Let (M³, g, k) be a complete asymptotically flat initial data set satisfying the dominant energy condition, and let m denote its ADM mass. The generalized Penrose conjecture asserts that the area of an outermost generalized apparent horizon N⊂ M satisfies |N| ≤ 16 πm². In this paper, we establish this inequality for each connected component of N in the case where k is proportional to the metric g. Our approach is based on a new geometric evolution, which we call the σ-inverse mean curvature flow, together with a novel monotonicity formula that may be of independent interest.

Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow

Tue, 19 May 2026 00:00:00 GMT

Mattia Fogagnolo, Giorgio Gatti, Alessandra Pluda — We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a C⁰ metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem for continuous Riemannian metrics with nonnegative scalar curvature in such IMCF sense.

Functional Renormalization Group as a Ricci Flow: An 𝓕-Entropy Perspective on Information Metric Dynamics

Sun, 17 May 2026 00:00:00 GMT

Jin Mo Bok, Ki-Seok Kim — We establish a geometric correspondence between the Functional Renormalization Group (FRG) and a Ricci flow modified by a potential-driven diffeomorphism. By rewriting the Polchinski exact RG equation as an infinite-dimensional Fokker–Planck equation for field-distribution functionals, we show how a probability flow driven by a "thermodynamic" free-energy functional induces the evolution of the Fisher information metric on the coupling-constant manifold. Using the continuous scale-dissipation rate of this free-energy functional, we construct an RG-flow entropy functional that provides an infinite-dimensional counterpart of Perelman's 𝓕-entropy. The parametric Hessian of this RG-flow entropy then encodes the scale deformation of the Fisher information metric, thereby linking the JKO–Wasserstein flow in field-configuration space to the geometry of the coupling-constant manifold. An emergent scalar information potential Φ encodes the potential-driven diffeomorphism component, restoring the tensorial form of the flow under reparametrizations of the coupling coordinates. In this representation, the successive integration of high-energy degrees of freedom effectively smooths out the curvature of the information manifold, so that RG fixed points are realized as steady Ricci soliton equilibria. These results connect quantum field theory, optimal transport, and Perelman's theory of geometric evolution, providing a geometric framework for characterizing the stability, universality, and topological structure of quantum field theories.

Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes

Thu, 14 May 2026 00:00:00 GMT

Tanay Gupta, Sukanta Panda, Rajib Saha — Recent cosmological tests have discovered a fresh new set of anomalies in the large-scale isotropy of the universe. Motivated thus by the numerous pieces of evidence for large-scale cosmic isotropy violation with the advent of the 'precision cosmology' era, we are led to explore the viability of anisotropic Thurston geometries, described in William Thurston's geometrization conjecture. In this work, we examine the coherent temperature and polarization signals generated in the CMB sky by such geometries. We begin with introducing Thurston spacetimes as our background model and the formalism we use to obtain the patterns. We then construct a set of transfer equations relative to a given background and solve them for each spacetime geometry. We finally discuss the role of spatial curvature in these FLRW limiting models along with their underlying geometry, and attempt to establish some general results on the symmetries of the patterns produced by their time evolution in terms of the Stokes parameters P, Q, U and V. We show the evolution of temperature and polarization amplitudes in terms of such Stokes parameters at different timestamps and attempt to isolate individual Thurston geometries.

The perturbative Ricci flow in gravity

Mon, 20 Apr 2026 00:00:00 GMT

Robert V. Harlander, Yannick Kluth, Jonas T. Kohnen, Henry Werthenbach — We develop a perturbative formulation of the Ricci flow in gravity. Following steps analogous to the gradient flow in QCD, we supplement the usual Feynman rules for perturbative gravity by flowed propagators and vertices as well as graviton flow lines which describe the evolution of gravity along the Ricci flow. By calculating vacuum expectation values of a number of independent operators at the two-loop level, we derive the required counterterms of the flowed action. Our results allow us to define a Ricci-flow based renormalization scheme for Newton's constant G_N. Studying its renormalization group behavior, we recover a non-Gaußian fixed point in accordance with well-known non-perturbative considerations

High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

Mon, 23 Feb 2026 00:00:00 GMT

Ben Andrews, Qiyu Zhou — In the pseudo-Euclidean space ℝ^(n+1,k), we consider the mean curvature flow of n-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension k. We show that if the initial submanifold is compact and spacelike-convex (the acceleration along every geodesic is strictly spacelike), then natural quantities measuring curvature pinching and noncollapsing are preserved under the flow. Moreover, we prove an analogue of the Huisken and Gage-Hamilton theorems in this setting, which states that the mean curvature flow deforms any such submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in a maximally spacelike affine subspace ℝ^(n+1,0)⊂ ℝ^(n+1,k).