We show that 3D gravity, in its pure connection formulation, admits a natural 6D interpretation. The 3D field equations for the connection are equivalent to 6D Hitchin equations for the Chern-Simons 3-form in the total space of the principal bundle over the 3-dimensional base. Turning this construction around one gets …
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48 results for “3D space forms”
The study characterizes constant curvature manifolds using ruled surfaces.
problem Characterizing manifolds of constant curvature using ruled surfaces.
method Investigating ruled surfaces in 3d Riemannian manifolds, finding stiction curve, distribution parameter, and fundamental forms.
result Identifies necessary and sufficient conditions for extrinsically flat surfaces to be ruled and proves manifold properties.
New method classifies HCMU surfaces in 3D space forms as Weingarten surfaces.
problem Classifying HCMU surfaces in 3D space forms as Weingarten surfaces.
method Totally different method from previous work.
result Criteria for Weingarten surfaces that are also HCMU surfaces.
Researchers classify 3D self-shrinkers in 4D space.
problem Classifying complete 3D self-shrinkers with specific properties in Euclidean space.
method Completely classified 3-dimensional complete self-shrinkers with constant norm of the second fundamental form and constant in .
result A complete classification of 3D self-shrinkers in Euclidean space .
Paper shows non-CSC HCMU metrics can't be isometrically immersed into 3D space forms.
problem Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
method Using moving frames to demonstrate the impossibility of isometric minimal immersion.
result Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
Study Seiberg-Witten moduli spaces on 3D cobordisms with Morse functions.
problem Properties of Seiberg-Witten moduli spaces on 3D cobordisms.
method Perturbed Seiberg-Witten equations on cylindrical ends with Morse functions.
result Properties of moduli spaces of Seiberg-Witten equations on 3D cobordisms.
Study complete 3D λ-translators in Minkowski space with constant properties.
problem Classify 3D space-like λ-translators with specific constant properties.
method Obtained classification theorem through analysis of constant norm and .
result Classification theorem for 3D complete space-like λ-translators.
Study on triharmonic curves in 3D spaces, proving their existence and classification.
problem Characterizing triharmonic curves in 3D homogeneous spaces.
method Analyzing curves with constant curvature in Riemannian manifolds, focusing on Frenet helices and space forms.
result Classification of triharmonic Frenet helices in space forms and Bianchi-Cartan-Vranceanu spaces.
Study classifies 3D self-shrinkers with constant second form norm.
problem Classifying self-shrinkers with specific geometric properties.
method Analyzes 3D self-shrinkers in Euclidean space with constant second form norm.
result Classifies complete self-shrinkers with constant norm of the second fundamental form.
Study 3d N=1 vacua from M-theory compactification on Spin(7) space.
problem Quantum corrections in 3d N=1 vacua from M-theory compactification.
method Use Higgs bundles to analyze 3d N=1 vacua and track corrections.
result Topological anomalies are robust and calculable in 3d effective field theory.
Study on 3D surfaces and tangles formed by Poncelet triangles.
problem Geometric and topological properties of 3D surfaces and tangles.
method Exploration of Poncelet triangles and associated points.
result Properties of 3D surfaces and tangles formed by Poncelet triangles.
The paper studies stable surfaces with constant curvature in 3D space forms.
problem Stability of surfaces with constant extrinsic curvature in space forms.
method Using stability notions for surfaces with constant higher order mean curvature.
result Rigidity results for surfaces with free boundary in geodesic balls or slabs.
The paper defines and analyzes conformal trajectories in 3D space forms.
problem Understanding trajectories in curved 3D spaces.
method Defined conformal trajectories and studied their properties in , , and .
result Conformal trajectories in and have constant curvature and torsion.
In 3D space forms, a lens minimizes volume for a fixed surface area.
problem Finding the shape with minimal volume for a given surface area in 3D space forms.
method Proving a sharp reverse isoperimetric inequality for -convex bodies.
result The -convex lens minimizes volume for a fixed surface area in 3D space forms.
In order to generate novel 3D shapes with machine learning, one must allow for interpolation. The typical approach for incorporating this creative process is to interpolate in a learned latent space so as to avoid the problem of generating unrealistic instances by exploiting the model's learned structure. The process o…
We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the spac…
This paper defines directional derivatives and solves Maxwell's equations in curved 3D space.
problem Analyzing electromagnetic fields in curved non-flat 3D space.
method Defined directional derivatives and used Frenet formulas to express Serret-Frenet relations. Solved Maxwell's equations for electric and magnetic fields.
result Solved Maxwell's equations for electromagnetic fields in curved 3D space.
Soft tilingsmath.MG
Softens tilings in 3D space, proving conjectures about polyhedral tilings.
problem Proving that all locally polyhedral tilings in 3D space can be softened.
method Developed a new edge-bending algorithm to prove the statement.
result Proved conjectures about polyhedral tilings in 3D space and the plane.
Geodesics spiral around Reeb orbits in 3D contact manifolds.
problem Understanding geodesics in sub-Riemannian geometry.
method Normal form along Reeb orbits due to Melrose.
result Sub-Riemannian geodesics spiral around Reeb orbits in both phase and configuration spaces.
The paper proves conjectures and classifies metrics on 3D manifolds.
problem Proving conjectures and classifying metrics on 3D manifolds with specific curvature conditions.
method Analytical proofs and classification theorems.
result Critical metrics on 3D manifolds are isometric to geodesic balls in space forms.
The paper finds formulas for special surface shapes in 3D space.
problem Creating formulas for constant mean curvature surfaces.
method Weierstrass representations for discrete surfaces in isotropic space.
result Constructs examples of surfaces with discrete parametrizations.
The paper classifies 3D self-expanders with specific properties.
problem Classifying self-expanders with constant curvature components.
method Complete classification of 3D self-expanders with specific curvature conditions.
result Completely classified 3D self-expanders with constant curvature components.
The study classifies biharmonic submersions from 3D BCV spaces.
problem Classifying biharmonic submersions from 3D BCV spaces.
method Complete classification through proving existence and constructing families of submersions.
result Proper biharmonic Riemannian submersions exist only in two specific cases.
3D contact manifolds have optimal higher systolic ratios.
problem Optimizing higher systolic ratios in 3D contact manifolds.
method Proving Besse contact forms maximize certain ratios.
result Besse contact forms are local maximizers of higher systolic ratios.
Standardizes surfaces in 3D handlebodies using Morse theory.
problem Classifying and identifying incompressible surfaces in 3D handlebodies.
method Applied Morse theory to provide a standard form and a condition for incompressibility.
result Developed a practical algorithm to determine incompressibility of surfaces in standard form.
3D adversarial logos can fool object detectors in real-world settings.
problem Creating robust adversarial attacks in 3D rendering views.
method Constructing 3D adversarial logos via texture mapping and differentiable rendering.
result 3D adversarial logos are more versatile and robust than traditional adversarial patches.
New proof of wave trace formula for 3D-contact manifolds.
problem Wave trace formula for 3D-contact manifolds.
method Normal form reduction to Heisenberg group.
result Extension of Chazarain-Duistermaat-Guillemin formula.
New method describes entanglement of straight lines in 3D space.
problem Tackles the geometry and topology of configurations of straight lines.
method Introduces direction matrices and a discrete motion principle.
result Shows n-crosses as links of pairwise connected unknots.
3d Modularityhep-th
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d theories where such structures a priori are not manifest. These modular structures include: mock modular forms, Weil representations, quantum mo…
Study of symplectic Monge-Ampère equations using moment maps and contact structures.
problem Characterizing symplectic Monge-Ampère equations through geometric structures.
method Constructing contact cone structures and using moment maps to relate equations to projective spaces.
result The contact cone structure and the cocharacteristic variety coincide for non-degenerate equations.
New form of -singularities for fronts in 3D space.
problem Understanding singularities of fronts in 3D space.
method Coordinate transformation on source and isometry on target.
result Computed differential geometric invariants near -singularity.
3D dual field theories for Virasoro minimal models constructed using Seifert fiber spaces.
problem Constructing 3D dual field theories for Virasoro minimal models.
method 3D-3D correspondence and Seifert fiber spaces.
result 3D dual field theories constructed for Virasoro minimal models.
Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-c…
Proves helicity is the only regular Casimir for 3D hydrodynamics.
problem Identifying unique Casimir functions for 3D hydrodynamics.
method Normal forms, Poincaré-Birkhoff theorem, division lemma.
result Helicity is the only regular Casimir for volume-preserving diffeomorphisms.
Classifies 3D manifolds with specific structures and automorphisms.
problem Classifying compact 3D manifolds with path structures and large automorphism groups.
method Uses Cartan connections and constant curvature analysis.
result Curvature of Cartan connections is constant for these manifolds.
Deep generative architectures provide a way to model not only images but also complex, 3-dimensional objects, such as point clouds. In this work, we present a novel method to obtain meaningful representations of 3D shapes that can be used for challenging tasks including 3D points generation, reconstruction, compression…
The theory of differential forms began with a discovery of Poincare who found conservation laws of a new type for Hamiltonian systems - The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant differential forms: a symplectic two-form, or a contact one-form for geodesic flo…
A Python implementation of some geometric tools on Kendall 3D shape space for practical applicationsstat.CO
Python tools for 3D shape analysis on Kendall's space.
problem Lack of practical utilities for advanced 3D shape analysis.
method Developed Python tools for 3D shape analysis on Kendall's 3D Shape Space.
result Efficient, accessible software solutions for researchers.
Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.
problem Developing equations for Dirac operators in complex 3D domains.
method First-kind boundary integral equations, generalized Garding inequalities, Fredholm operators, finite dimensional kernels, Betti numbers.
result Finite dimensional kernels equal to the sum of Betti numbers, explaining the bilinear forms.
Holomorphic functions from knot complements link to quantum modular forms.
problem Analyzing holomorphic functions from knot complements.
method Matrix-valued holomorphic functions, cocycles, and quantum modularity.
result Identifies a matrix-valued holomorphic quantum modular form.
New method reconstructs 3D shapes from 2D images using Kendall's shape space.
problem Reconstruct 3D shapes from 2D images, especially for rare specimens.
method Kendall's shape space approach with prior information.
result More robust and plausible shapes compared to previous methods.
Study helical motions of lines in 3D spaces, solving control problems.
problem Controlling helical motions of lines in 3D spaces.
method Analyzing control systems on manifolds of oriented geodesics in 3D spaces of different curvatures.
result The system is controllable if and only if alpha^2 ≠ kappa.
Deep learning models predict option prices from 3D tensor data.
problem Predicting option prices for risk management and trading.
method 3D tensor representation of financial data, deep learning models (2D tensors in 3 channels).
result Proposed models outperform traditional methods like B-S model and vector-based LSTM.
Non-trivialization probability of arc system in 3D space
problem Defining and generalizing the knotting probability of an arc diagram in 3D space
method Transforming polygonal arcs in 3D space into unique arc diagrams
result Introducing and generalizing the Non-Trivialization probability (NT probability) for arc systems in 3D space
Study null curves and their motion in 3D flat space-time, leading to integrable hierarchies.
problem Understanding null curves and their motion in 3D flat space-time.
method Analyzing the motion of null curves and their surfaces, deriving integrability conditions and hierarchies.
result Obtained one- and two-soliton surfaces associated with the MKdV equation, showing singularities in finite time.
Study of asymptotics of meromorphic 3D-index as q approaches 1.
problem Understanding the asymptotic behavior of a meromorphic function related to 3D-index.
method Developed a conjectural asymptotic approximation using stationary phase analysis of a circle-valued angle structure integral.
result Found connections to angle structures and volume optimization.
NT probability measures knotting in 3D arc systems.
problem Measuring knotting in 3D arc systems.
method Transforming polygonal arcs into unique diagrams, generalizing NT probability.
result Properties of NT probability for 3D arc systems are shown.
Proposes a neural network for recognizing 3D skeleton-based interactions.
problem Recognizing two-person interactions from 3D skeleton sequences.
method Uses Gaussian distributions and Riemannian geometry of SPD matrices and matrix groups.
result Achieves competitive results on three benchmarks for 3D human activity understanding.