The Penrose-Kauffman polynomial connects knot theory to graph coloring.
problem Understanding the Penrose-Kauffman polynomial for cubic graphs.
method Using knot theory, the polynomial is shown equivalent to 3-coloring link diagrams.
result The Four Color Theorem is linked to 3-coloring link diagrams.
The paper explores unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
problem Exploring Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
method Using the Penrose diagram for conformal compactification, the paper investigates unique properties of Darboux transformations of spacelike curves.
result The paper identifies unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane, especially regarding singularities and blowup.
New invariant for special alternating links based on graph Laplacian.
problem Developing an invariant for special alternating links.
method Using the Laplacian matrix of the Tait graph, invariant is defined.
result A specific quadratic trace expression is invariant under flype moves.
In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These \emph{virtual graphs} are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, t…
Differential Cohomotopy theory predicts brane interactions via chord diagrams.
problem Quantization of brane charges and moduli spaces.
method Differential refinement of Cohomotopy theory, configuration spaces, chord diagrams.
result Higher observables on brane moduli spaces are given by weight systems on chord diagrams.
Polynomials derived from Heegaard diagrams for 3-manifolds.
problem Creating polynomial invariants for 3-manifolds.
method Using Heegaard diagrams and cellularly embedded graphs to define polynomials.
result Finiteness of minimal Heegaard graphs for a fixed manifold enables polynomial definition.
We analyze the horizon and geodesic structure of a class of 4D off--diagonal metrics with deformed spherical symmetries, which are exact solutions of the vacuum Einstein equations with anholonomic variables. The maximal analytic extension of the ellipsoid type metrics are constructed and the Penrose diagrams are analyz…
Counterexample disproves recent Penrose conjecture variant.
problem A variant of the Penrose conjecture.
method Provided a counterexample.
result The conjectured variant of the Penrose inequality is false.
Constructs homologies for ribbon graphs to recover Penrose polynomials.
problem Recovering Penrose polynomials from ribbon graphs.
method TQFT-valued cube of resolutions from two-dimensional cobordisms.
result Euler characteristics of constructed homologies recover Penrose polynomials.
New relations for Penrose polynomial at n=4 and n=3.
problem Evaluation of Penrose polynomial at specific values of n.
method New ribbon graph polynomial and extended relations for n=3 to all n.
result Proven new relations for Penrose polynomial at n=4 and extended relations for n=3.
Study how past eon's matter affects present eon in Penrose's cyclic cosmology.
problem Determining present eon's matter content from past eon's matter.
method Use Penrose's reciprocity hypothesis to link past and present eons' matter.
result Perfect fluid matter content of past eon influences present eon's matter content.
Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
problem Stability of the Schwarzschild 3-manifold in the context of the 3D Riemannian Penrose inequality.
method Pointed measured Gromov-Hausdorff topology, negligible domains and boundary area perturbations.
result Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
Smooth metrics satisfying Penrose inequality are necessarily smooth.
problem Rigidity of Penrose inequality with singular metrics.
method Showed suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality are smooth in specified coordinates.
result Smooth metrics satisfying Penrose inequality are necessarily smooth.
We investigate the Penrose limits of classical string and M-theory backgrounds. We prove that the number of (super)symmetries of a supergravity background never decreases in the limit. We classify all the possible Penrose limits of AdS x S spacetimes and of supergravity brane solutions. We also present the Penrose limi…
Extends Penrose limit to Finsler spacetimes.
problem Extending Penrose limit to Finsler spacetimes.
method Introducing lightlike coordinates and adapting Lorentzian pp-wave definition.
result New examples of Finsler pp-waves presented.
Penrose limit results for specific 3-surfaces in space-time.
problem Understanding Penrose limits for specific geometric configurations.
method Analyzing Penrose limits along null geodesics in umbilic 3-surfaces.
result Penrose limit of a specific 3-surface results in a diagonalisable plane-wave.
Throughout the literature on the charged Riemannian Penrose inequality, it is generally assumed that there is no charged matter present; that is, the electric field is divergence-free. The aim of this article is to clarify when the charged Riemannian Penrose inequality holds in the presence of charged matter, and when …
Paper applies Newman-Penrose formalism to ACM manifolds.
problem Classifying compact ACM manifolds with η-Einstein metrics. method Newman-Penrose formalism applied to ACM manifolds.
result Classification of compact ACM manifolds with η-Einstein metrics. New proof of Penrose inequality using potential theory.
problem Proving the Riemannian Penrose inequality for black holes.
method Establishing a monotonicity formula for the p-capacitary potential.
result A new proof of the Penrose inequality for black holes.
Conditions for Penrose-Ward transformation on specific manifolds.
problem Conditions for Penrose-Ward transformation on almost G2-manifolds with almost twistorial structures. method Necessary and sufficient conditions derived through Penrose-Ward transformation.
result Conditions for Penrose-Ward transformation on almost G2-manifolds with almost twistorial structures. We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We prove that the inequality holds in several important cases.
Researchers prove charged Penrose inequality and positive mass theorem for specific manifold types.
problem Proving inequalities for charged initial data sets with cylindrical ends.
method Doubling argument and application of existing results by Weinstein, Yamada, and Khuri, Weinstein, Yamada.
result Established charged Penrose inequality and positive mass theorem for time symmetric initial data sets with cylindrical ends.
The null Penrose inequality, i.e. the Penrose inequality in terms of the Bondi energy, is studied by introducing a funtional on surfaces and studying its properties along a null hypersurface Ω extending to past null infinity. We prove a general Penrose-type inequality which involves the limit at infinity of the Hawki…
The article proves charged quasi-local Penrose inequalities for compact manifolds with boundary.
problem Determining if a quasi-local version of the Riemannian Penrose inequality holds for Einstein-Maxwell equations.
method Building on ideas of Lu and Miao, and the first-named author, the article proves charged quasi-local Penrose inequalities for a class of compact manifolds with boundary.
result The lower bound on quasi-local mass is exactly the lower bound on the ADM mass given by the charged Riemannian Penrose inequality for a specific reference manifold.
The purpose of this article is to view the Penrose kite from the perspective of symplectic geometry.
Proves existence of solutions to Einstein constraints with specific boundary conditions and verifies Penrose inequality.
problem Existence of asymptotically hyperbolic solutions to Einstein constraints with marginally outer trapped boundaries.
method Constant mean curvature conformal method.
result Verification of Penrose inequality for certain Schwarzschild-AdS black hole perturbations.
The conformal flow of metrics [2] has been used to successfully establish a special case of the Penrose inequality, which yields a lower bound for the total mass of a spacetime in terms of horizon area. Here we show how to adapt the conformal flow of metrics, so that it may be applied to the Penrose inequality for gene…
Establishes a Penrose-type inequality for static spacetimes.
problem Finding a lower bound on the total mass of static spacetimes.
method Analyzes (n+1)-dimensional asymptotically flat standard static spacetimes under timelike convergence condition.
result Extends Penrose-type inequalities to all dimensions and characterizes equality conditions.
Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.
problem Understand the global structure of spacetimes with weakly trapped surfaces.
method Show foliation of MOTS generating totally geodesic null hypersurfaces.
result Obtain local or global rigidity results based on assumptions.
We establish versions of the Positive Mass and Penrose inequalities for a class of asymptotically hyperbolic hypersurfaces. In particular, under the usual dominant energy condition, we prove in all dimensions n≥3 an optimal Penrose inequality for certain graphs in hyperbolic space Hn+1 whose boundary…
Proves Penrose inequality in all dimensions for specific manifolds.
problem Proving Penrose inequality in arbitrary dimensions for certain manifolds.
method Extends Bray's conformal-flow method to higher dimensions, dealing with singular outer-minimizing enclosures.
result Proves the Riemannian Penrose inequality in arbitrary dimensions.
Penrose conjecture proven for specific initial data sets.
problem Proving Penrose conjecture for certain types of initial data sets.
method Used σ-inverse mean curvature flow and a monotonicity formula.
result Penrose conjecture established for 2-convex initial data sets.
Various complexes of differential operators are constructed on complex projective space via the Penrose transform, which also computes their cohomology.
Proves Riemannian Penrose Inequality for specific manifolds.
problem Proving Riemannian Penrose Inequality for certain manifolds.
method Novel interplay between Hawking mass and potential-theoretic Hawking mass.
result Establishes equality between ADM mass and Huisken's Isoperimetric mass.
We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen--Wallach space. We then consider several homogenous examples to show that these results are indeed sha…
In arXiv:0905.2622v1 and arXiv:0910.4785v1, Bray and Khuri outlined an approach to prove the Penrose inequality for general initial data sets of the Einstein equations. In this paper we extend this approach so that it may be applied to a charged version of the Penrose inequality. Moreover, assuming that the initial dat…
Study Penrose inequality for metrics with singular sets.
problem Penrose inequality for metrics with singular sets.
method Analysis of Penrose inequality for metrics with singular sets of dimension less than n-1, without additional conditions.
result Complement existing results by studying Penrose inequality for metrics with singular sets of lower dimension.
Current high-throughput data acquisition technologies probe dynamical systems with different imaging modalities, generating massive data sets at different spatial and temporal resolutions posing challenging problems in multimodal data fusion. A case in point is the attempt to parse out the brain structures and networks…
The Penrose inequality gives a lower bound for the total mass of a spacetime in terms of the area of suitable surfaces that represent black holes. Its validity is supported by the cosmic censorship conjecture and therefore its proof (or disproof) is an important problem in relation with gravitational collapse. The Penr…
We point out that algebraically special Einstein fields with twisting rays exhibit the basic properties of conformal Universes considered recently by Roger Penrose.
The positive mass theorem is one of the fundamental results in general relativity. It states that, assuming the dominant energy condition, the total mass of an asymptotically flat spacetime is non-negative. The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the…
Higher-dimensional Schwarzschild spacetimes violate the Penrose property.
problem Causal behavior of higher-dimensional Schwarzschild spacetimes.
method Analyzing causal properties in (2+1), (3+1), and (d+1) dimensions. result The Penrose property does not hold for (d+1) dimensional Schwarzschild if d>3. We study the Penrose transform for the `quaternionic objects' whose twistor spaces are complex manifolds endowed with locally complete families of embedded Riemann spheres with positive normal bundles.
In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool …
The paper proves a Penrose inequality for 4-discs with hyperbolic ends and conic singularities.
problem Proving a Penrose inequality for conformal metrics on a unit 4-disc with hyperbolic ends and conic singularities.
method Defining a mass term and proving a Penrose type inequality with curvature condition.
result Proves a Penrose inequality for conformal metrics on a unit 4-disc with hyperbolic ends and conic singularities.
Researchers prove a Penrose inequality for spacetime with specific conditions.
problem Establishing mass lower bounds for spacetime with specific asymptotic conditions.
method Combining harmonic level set approach, Jang equation, and stability techniques.
result Proof of Penrose inequality with universal constant and minimal area requirement.
Geometrically studies Moore-Penrose inverse and polar decomposition continuity.
problem Continuity of Moore-Penrose inverse for perturbations by operator ideals.
method Geometric construction using essential codimension and Banach-Lie group action.
result Moore-Penrose inverse is a real analytic map between manifolds.
Paper proves a Penrose inequality in extrinsic geometry.
problem Proving a Penrose inequality in extrinsic geometry.
method Analyzing minimal capillary surfaces and their free energy.
result Established an extrinsic Penrose inequality.