HR in 8D encodes unique conformal gravity with negative curvature.
problem Holographic Renormalisation in 8D Einstein Gravity.
method Relating HR to Topological Regularisation and adding the Euler term.
result The unique conformal gravity theory reproduces the polynomial and cancels divergent terms.
The paper finds at least four prime closed characteristics on star-shaped hypersurfaces in 8D space.
problem Finding prime closed characteristics on compact star-shaped hypersurfaces in 8D space.
method Proved existence of at least four prime closed characteristics for non-degenerate C3 compact star-shaped hypersurfaces in R8 without prime closed characteristics of Maslov-type index -1. result Existence of at least four prime closed characteristics on compact star-shaped hypersurfaces in R8. Study on harmonic flow of Spin(7)-structures in 8D manifolds.
problem Comparing isometric Spin(7)-structures.
method Developed a notion of harmonicity for Spin(7)-structures and studied the harmonic flow.
result Analytical properties of the harmonic flow of Spin(7)-structures presented.
Novel gauge-theoretic Floer homologies defined for 7, 6, and 5-manifolds.
problem Defining novel gauge-theoretic Floer homologies for different dimensions.
method Topologically-twisted 8d N=1 gauge theory on Spin(7)-manifolds. result Derivation of novel Floer homologies and A∞-categories. We simplify and extend a 6D conformal gravity theory to 8D, linking it to Q-curvature.
problem Constructing and understanding conformal gravity actions in different dimensions.
method Streamlined construction of 6D action, proving existence of 8D action, relating to Q-curvature.
result A unique 8D conformal gravity action exists with Einstein metrics as solutions.
The paper studies 8D manifolds with a specific tensor field called a cubic discriminant.
problem Characterizing and understanding 8D Riemannian manifolds with reduced structure groups.
method Introducing an almost quaternion-Hermitian structure and a cubic discriminant tensor field.
result Only two non-flat, integrable examples of these structures are found: quaternion-Kähler symmetric spaces.
New proof of a unique 3-part partition in 8D space.
problem Existence of a non-standard isoperimetric partition in high dimensions.
method Analytical proof showing existence of a specific partition.
result Existence of a non-standard isoperimetric partition in 8D space.
DM approximates submanifolds with error bounds.
problem Understanding the accuracy of Diffusion Maps in embedding submanifolds.
method Deriving geometric properties and deriving bounds on embedding errors.
result Error bounds for DM embeddings and tangent spaces.
Generic smooth boundaries for isoperimetric regions in 8D manifolds.
problem Understanding boundaries of isoperimetric regions in high-dimensional spaces.
method Generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight.
result Smooth nondegenerate boundaries for isoperimetric regions for generic metrics and volumes.
Minimal surfaces in 8D smooth and nondegenerate.
problem Generic regularity of minimal hypersurfaces in 8D.
method Analysis of C∞-generic metrics. result All minimal hypersurfaces are smooth and nondegenerate.
Generic smooth minimal hypersurfaces exist in 8D manifolds.
problem Existence of smooth minimal hypersurfaces in high-dimensional manifolds.
method Global perturbation argument and a novel geometric invariant.
result Generic metrics on 8D manifolds admit smooth minimal hypersurfaces.
Novel A∞-categories derived from gauge theories for manifold homologies.
problem Categorifying manifold homologies via gauge theories.
method 3d and 8d gauged Landau-Ginzburg models, higher A∞-categories. result Derived novel A∞-categories for various manifold homologies. Researchers found a new 8D Taub-NUT-like metric using harmonic superspace.
problem Finding a new 8-dimensional Taub-NUT-like manifold.
method Used harmonic superspace formalism to derive the metric.
result Derived a new 8D Taub-NUT-like metric.
New bounds on shortest geodesic loops on a sphere.
problem Finding shortest geodesic loops on a sphere.
method Analyzing geodesic loops starting and ending at a fixed point on a sphere.
result At any point on a sphere, there are at least two distinct geodesic loops whose lengths are bounded by 8d and 14d.
This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Ampère equations, by using the so-called "meta-symplectic structure" ass…
Formula derived for Spin(7)-structures on 8D manifolds.
problem Deriving the intrinsic torsion for Spin(7)-structures.
method Explicit formula derivation using Levi-Civita and minimal Spin(7)-connections.
result Formula relating intrinsic torsion to Lee form and codifferential components.
Study classifies 8D Lie groups with specific foliations and Hermitian structures.
problem Classifying 8D Lie groups with specific conformal and minimal foliations.
method Investigates left-invariant Hermitian structures on SU(2)×SU(2) leaves of 8D Lie groups. result Classifies Lie groups based on the integrability and types of Hermitian structures.
New G2 symmetry found in 8D distribution with 6D square.
problem Discovering new G2 symmetry in geometric distributions. method Analyzing rank 3 distribution on 8D manifold with growth vector (3,6,8).
result Maximally symmetric rank 3 distribution with 6D square.