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Relatively little is known about the discrete horospheres in hyperbolic groups, even in simple settings. In this paper we work with hyperbolic one-ended right-angled Coxeter groups and describe two graph structures that mimic the intrinsic metric on a classical horosphere: the Rips graph and the divergence graph (the latter due to Cohen, Goodman-Strauss, and Rieck). We develop, analyze, and implement algorithms based on finite-state machines that draw large finite portions of these graphs, and deduce various geometric corollaries about the path metrics induced by these graph structures.
We consider the existence of cohomogeneity one solitons for the isometric flow of $G_2$-structures on the following classes of torsion-free $G_2$-manifolds: the Euclidean $R^7$ with its standard $G_2$-structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon $G_2$-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on $R^7$ is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.
We determine the Néron-Severi lattices of $K3$ hypersurfaces with large Picard number in toric three-folds derived from Fano polytopes. On each $K3$ surface, we introduce a particular elliptic fibration. In the proof of the main theorem, we show that the Néron-Severi lattice of each $K3$ surface is generated by a general fibre, sections and appropriately selected components of the singular fibres of our elliptic fibration. Our argument gives a certain proof of the Dolgachev conjecture for Fano polytopes, which is a conjecture on mirror symmetry for $K3$ surfaces.
These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with these notes in a forthcoming issue of Geometriae Dedicata.
We construct complete Calabi-Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection $V_0$ that is a Calabi-Yau cone, extending the work of Székelyhidi (2019). The constructed Calabi-Yau manifold has tangent cone at infinity given by $\mathbb{C} \times V_0$. This construction produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.
Given a map f: M \to M of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S^1, these torsion obstructions are identified with the ones due to Farrell. We have changed the exposition according to the comments of the referee and corrected some typos. The paper will appear in Geometriae Dedicata.
Researchers have proposed that black holes stop evaporating at the last moment, leaving behind tiny remnants that preserve all the information they contain。 The same seven-dimensional geometry behind this idea could also help explain why elementary particles have mass
The Earth may not be that massive, but it still distorts space-time
Researchers have created quantum control techniques that can make a system appear to run backward in time。 By precisely managing quantum measurements, they can reshape the system's arrow of time and even harvest energy from the measurement process itself。 The breakthrough could lead to more powerful quantum computers, quantum batteries, and other a
Astronomers have released the largest gravitational wave catalog ever, revealing 161 new black hole collisions and pushing the total number of detections to 390。 Among the highlights are the clearest gravitational wave signal ever recorded, the most accurate location of a black hole merger, and growing evidence that some black holes are the product
"I have been waiting a long time to finally get up there
A major breakthrough in quantum technology has turned magnons, tiny magnetic waves once considered too short-lived for practical use, into promising carriers of quantum information。 Researchers extended their lifetime by nearly 100 times, reaching up to 18 microseconds, and discovered that the main limitation is not a law of physics but the purity
Rebranded Codex promises independent workflows that can run "for hours if needed
Two newly confirmed "super-puff" planets are so diffuse that they are less dense than cotton candy, despite being about the size of Jupiter。 Their rare orbital relationship and enormous, lightweight atmospheres could provide valuable clues about how some of the strangest planets in the galaxy come to exist
A planet with one side permanently roasting and the other frozen in endless darkness might still have a chance of supporting life。 Researchers found that heat inside a tidally locked exoplanet could circulate in a stable, continuous loop, helping moderate temperatures in certain regions。 Their laboratory model suggests these worlds may be more hosp
NASA’s Lucy spacecraft discovered that asteroid Donaldjohanson is a wobbling, peanut-shaped relic born from a violent collision and slowly reshaped by the subtle force of sunlight。 It also carries traces of ancient water, making it an important clue to the solar system’s mysterious past
Scientists have rewritten the story of gallium after discovering that its unusual atomic bonds re-form at high temperatures, contradicting decades of accepted theory。 The finding changes how researchers explain why the metal melts so easily and behaves unlike almost any other metal。 Beyond solving a long-standing scientific mystery, the work could
A new sunlight-powered material can convert visible light into higher-energy UV light, overcoming a challenge that has frustrated scientists for years。 The breakthrough could enable cleaner air purification, solar-driven chemistry, and advanced manufacturing technologies using nothing more than natural sunlight
Scientists have identified new clues that could help astronomers spot one of the most famous hypothetical alien megastructures: a Dyson sphere。 The study finds that red dwarfs and white dwarfs are the most promising stars to examine, since advanced civilizations could potentially build energy-harvesting swarms around them more easily。 These objects
A new quantum device can generate precisely controlled bursts of sound-like particles, or phonons, by forcing electrons through an ultra-thin crystal at extremely low temperatures。 The surprising behavior pushes beyond the limits predicted by current theories, suggesting scientists need to rethink how energy moves through advanced materials。 In the