The paper offers new ways to solve map coloring problems.
problem Coloring plane maps with four colors.
method Alternate formulations including a tautological expansion and an extended Penrose Bracket.
result Extended Penrose Bracket counts colorings of arbitrary cubic graphs.
By generalizing the Kuperberg sl(3) bracket, we construct a graph-valued analogue of the Homflypt sl(3) invariant for virtual knots. The restriction of this invariant for classical knots coincides with the usual Homflypt sl(3) invariant, and for virtual knots and graphs it provides new information that allows one to pr…
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.
This paper examines the charged Riemannian Penrose inequality with and without charged matter.
problem When does the charged Riemannian Penrose inequality hold with charged matter?
method Revisited Jang's proof and constructed counterexamples to explore conditions for the inequality.
result Charged Riemannian Penrose inequality holds with suitable conditions on charged matter, but requires charge density not changing sign.
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 construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allo…
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…
New mass functional proves Penrose conjecture.
problem Null Penrose conjecture in general relativity.
method Defined and used a new quasi-local mass functional.
result Proved the null Penrose conjecture.
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.
Study on null Penrose inequality using geodesic foliations and functional properties.
problem Proving the null Penrose inequality using functional properties and geodesic foliations.
method Introducing a functional on surfaces, studying its properties along a null hypersurface, and using geodesic foliations to derive inequalities.
result Derivation of a general Penrose-type inequality and conditions for the null Penrose inequality.
Proves a new Penrose inequality for static spacetimes.
problem Establishing a lower bound on mass using area and static references.
method Develops a new quasi-local Penrose inequality for spherically symmetric static spacetimes.
result Proves a quasi-local Penrose inequality for any spherically symmetric static spacetime.
Penrose inequality proven for flat manifolds in dimensions less than 8.
problem Proving Penrose inequality for flat manifolds in specific dimensions.
method Outer minimizing boundary, nonnegative scalar curvature, mean curvature, scalar curvature.
result Penrose inequality proven for asymptotically flat manifolds in dimensions less than 8.
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. Paper proves rigidity for Penrose inequality on 3-manifolds with corners.
problem Proving rigidity for Penrose inequality on specific 3-manifolds.
method Analyzes asymptotically flat manifolds with nonnegative scalar curvature and corners.
result Proves rigidity for equality cases of Penrose inequality.
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.
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.
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.
Paper proves Penrose inequality for graphs over specific spacetime manifolds.
problem Proving Penrose inequality for graphs over Reissner-Nordström-anti-deSitter manifold.
method Inverse mean curvature flow to establish optimal Minkowski type inequality and Penrose type inequality.
result Established Penrose type inequality for graphs over Reissner-Nordström-anti-deSitter 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.
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.
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 for specific asymptotically flat manifolds.
problem Proving Penrose inequality for certain types of manifolds.
method Developed a new approximation scheme for a flow and established monotonicity of a free boundary Hawking mass.
result Proved the Riemannian Penrose inequality for specified manifolds.
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.
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.
Various complexes of differential operators are constructed on complex projective space via the Penrose transform, which also computes their cohomology.
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…
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…
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.
The paper proves the Null Penrose Inequality for Schwarzschild spacetimes.
problem Proving the Null Penrose Inequality for Schwarzschild spacetimes.
method Introducing quasi-round MOTSs and showing their stability and existence for specific horizons.
result Identifies sufficient conditions for the Null Penrose Inequality in Schwarzschild spacetimes.
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.
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. 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 …
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.
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.