Study constructs disks with curved boundaries in a 3D ball.
problem Constructing non-planar free boundary disks in a unit ball.
method Infinite family of non-planar disks with non-positive Gaussian curvature.
result Constructs disks with curved boundaries in a unit ball.
No free boundary Möbius bands exist in a 3D ball.
problem Proving the non-existence of Möbius bands with free boundaries in a 3D ball.
method Analytical proof based on geometric properties.
result Proves the non-existence of free boundary Möbius bands in the unit three-ball.
We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure
The paper studies minimal surfaces in 3D spheres and balls, confirming conjectures and identifying new surfaces.
problem Understanding minimal surfaces in 3D spheres and balls with low area.
method Equivariant optimization of Laplace and Steklov eigenvalues to construct minimal surfaces of prescribed topology.
result Sharp area estimates and varifold limits for minimal surfaces in large topology regimes.
The study shows a 3D manifold's macroscopic dimension is 1 under specific curvature constraints.
problem Understanding the macroscopic dimension of 3D Riemannian manifolds with curvature restrictions.
method Analyzing the volume and homology of balls in the manifold.
result A 3D manifold with the specified curvature constraints has macroscopic dimension 1.
New method speeds up knot computations in 3D.
problem Computational complexity in knot theory.
method 3D representation of knots for faster computation.
result Savings in computational complexity for knot invariants.
Study sub-Riemannian geodesics on a Heisenberg 3D nil-manifold.
problem Characterize geodesics on a specific nil-manifold.
method Analyze the projection of sub-Riemannian structure, describe geodesic flow dynamics, and estimate sub-Riemannian geodesics.
result Sharp bounds and estimates for sub-Riemannian geodesics.
Existing networks directly learn feature representations on 3D point clouds for shape analysis. We argue that 3D point clouds are highly redundant and hold irregular (permutation-invariant) structure, which makes it difficult to achieve inter-class discrimination efficiently. In this paper, we propose a two-faceted sol…
New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.
problem Existence of non-isotopic handlebodies with identical boundary.
method Construction of specific genus-g handlebodies in S4 and B5. result Proves Budney-Gabai conjecture for genus at least 2.
This paper embeds surfaces in 3D spheres and balls with minimal area.
problem Embed surfaces with boundary in B3 as minimal surfaces. method Optimizing Laplace and Steklov eigenvalues with symmetry groups.
result Proves existence of minimal surfaces in B3 with area below 2π. Convolution is an efficient technique to obtain abstract feature representations using hierarchical layers in deep networks. Although performing convolution in Euclidean geometries is fairly straightforward, its extension to other topological spaces---such as a sphere (S2) or a unit ball (B3)---…
Mogami introduced in 1995 a large class of triangulated 3-dimensional pseudomanifolds, henceforth called "Mogami pseudomanifolds". He proved an exponential bound for the size of this class in terms of the number of tetrahedra. The question of whether all 3-balls are Mogami has remained open since, a positive answer wou…
Solves Brezis' first open problem on ball solutions.
problem Existence of solutions to Brezis-Nirenberg problem on a 3D ball.
method Building on sign-changing solutions to the Yamabe problem.
result Infinitely many sign-changing, nonradial solutions found.
Refines knot defect measurement in 3D and 4D.
problem Measuring how far knots are from being alternating.
method Extends spanning surface defect to 4-ball, making comparisons and proving formulas.
result Connected sum formula proven.
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.
New theorem shows shapes close to balls, flow converges to balls in 2D and 3D.
problem Understanding the asymptotic behavior of volume-preserving mean curvature flow.
method Proved a new quantitative Alexandrov theorem and used it to show flow convergence.
result Weak solutions of volume-preserving mean curvature flow converge to disjoint balls in R^2 and R^3.
A new method calculates the minimum volume swept by a sphere's homotopy in 3D space.
problem Finding the minimum volume swept by a sphere's homotopy in 3D space.
method Cable system approach to define and compute cable indices.
result A linear-time algorithm computes all cable indices and achieves the lower bound for the swept volume.
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 study bounds the topology of free boundary minimal surfaces in 3D manifolds.
problem Understanding the topology of free boundary minimal surfaces in compact 3D manifolds.
method Establishing general bounds on the topology via min-max methods and analyzing varifolds.
result The first Betti number is lower semicontinuous in the limit of min-max sequences.
New shapes enclose less volume than the sphere, surprising in 3D.
problem Finding the minimal volume enclosed by smooth spheres with bounded curvatures.
method Produced a family of bodies parameterized by ε, each bounded by a smooth topological sphere with principal curvatures in [-1, 1].
result The unit sphere does not enclose the minimal volume among all smooth spheres in R^3 with principal curvatures in [-1, 1].
Self-affine tiles homeomorphic to a ball proven for a specific digit set.
problem Topology of self-affine tiles with collinear digit sets.
method Proving homeomorphism to a ball using integral self-affine tiles with collinear digit sets.
result A large class of integral self-affine tiles with collinear digit sets is homeomorphic to a closed 3-dimensional ball.
The study examines vector fields with integer singularities in 3D balls.
problem Characterizing the strong Lp-closure of vector fields with finitely many integer singularities. method Characterization and decomposition of vector fields with finitely many integer singularities.
result Decomposition theorem for elements in LZ1(B), revealing information about mass-minimizing currents. Smooth analog of Gromov's dihedral rigidity for 3D weakly convex domains.
problem Rigidity of 3D weakly convex domains with nonnegative scalar curvature.
method Capillary minimal surfaces and foliations with nonnegative mean curvature.
result Smooth analog of Gromov's dihedral rigidity for 3D weakly convex domains.
The paper proves a new inequality linking mass and volume in 3D space.
problem Relating mass and volume in 3D space with a sharp inequality.
method Using a monotonicity formula for level sets of a 3-harmonic function.
result Sharp lower bound for ADM mass in terms of Euclidean volume of Ω.
Study on Vapnik-Chervonenkis dimension of product intervals in R^d.
problem Combinatorial complexity of product intervals in R^d.
method Vapnik-Chervonenkis geometry approach.
result Vapnik-Chervonenkis dimension of balls in ℓ∞^d equals (3d+1)/2.
Proves uniqueness of capillary disks in 3D domains.
problem Proving uniqueness of capillary disks in three-dimensional domains modeled by elliptic PDEs.
method Using elliptic PDEs and properties of surfaces in 3D domains, generalizing Nitsche's and Hopf's theorems.
result Generalizes Nitsche's result for capillary constant mean curvature disks in the Euclidean ball.
Study on minimal surfaces with closed curvature lines in 3D space.
problem Investigating complete non-orientable minimal surfaces with specific curvature properties.
method Analyzing complete non-orientable minimal surfaces of finite total curvature in R3 with ends foliated by closed lines of curvature. result There are no such surfaces with one end, proving a rigid situation.
Study finds minimal hypersurfaces grow linearly in index, contrary to 3D.
problem Understanding index growth of minimal hypersurfaces.
method Partitioning methods for compact Lie groups, applied to hypersurfaces.
result Linear index growth for hypersurfaces of fixed topological type.
The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensional Ricci flow, and leads to local C/t deca…
Study of loop braid groups for 3D manifolds, linking algebra and dynamics.
problem Lack of a 3D framework for braid group theory in topological dynamics.
method Introduce loop braid groups and associate Burau matrix representations with generalized Lefschetz number.
result Established a connection between loop braid groups and topological dynamical properties, providing estimates for periodic points.
The paper disproves a conjecture about 3D manifolds using even lattice points.
problem Thurston's Euler class one conjecture for fillable contact structures.
method Analyzing finite covers of hyperbolic 3-manifolds and properties of their dual Thurston norm unit balls.
result Found counter-examples to the conjecture using even lattice points on boundary.
Higher surgeries preserve Steklov spectra in 3D and above.
problem Effect of topology changes on Steklov eigenvalues in higher dimensions.
method Perform surgeries of codimension 2 or higher on compact manifolds.
result Topology changes do not affect Steklov spectra in dimensions 3 and above.
We study knots in 3d Chern-Simons theory with complex gauge group SL(N,C), in the context of its relation with 3d N=2 theory (the so-called 3d-3d correspondence). The defect has either co-dimension 2 or co-dimension 4 inside the 6d (2,0) theory, which is compactified on a 3-manifold M^. …
Study of 3d-3d correspondence involving q-Weyl algebra and 3d-index.
problem Understanding the action of a q-Weyl algebra on the 3d-index of knots. method Investigation of the q-Weyl algebra's module action on the 3d-index, conjecturing structural properties. result Bilinear factorization, pair of linear q-difference equations, and rational function matrix for the 3d-index determination. 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.
3D flying wings created for any angle asymptotic cones.
problem Creating 3D steady gradient Ricci solitons with any angle asymptotic cones.
method Constructing 3D flying wings for any angle asymptotic cones.
result 3D flying wings constructed for any angle asymptotic cones.
Proposes a new effective central charge for 3d N=2 theories.
problem Understanding the effective central charge in 3d N=2 theories.
method Analyzes the superconformal index to propose a new quantity and discusses its properties and computation.
result Proposes a new effective central charge for 3d N=2 theories.
Autonomous driving requires 3D perception of vehicles and other objects in the in environment. Much of the current methods support 2D vehicle detection. This paper proposes a flexible pipeline to adopt any 2D detection network and fuse it with a 3D point cloud to generate 3D information with minimum changes of the 2D d…
Smooth 3D flows from non-smooth starting points.
problem Creating smooth Ricci flows from non-smooth initial conditions.
method Generalized singular Ricci flow applied to 3D complete manifolds.
result Existence of smooth Ricci flows starting from non-smooth initial conditions.
3D Axial-Attention improves lung nodule classification accuracy.
problem Limited 3D attention in existing methods.
method Proposes 3D Axial-Attention network with 3D positional encoding.
result 3D Axial-Attention achieves state-of-the-art performance.
The paper quantifies how scalar curvature changes under C0 convergence in 3D.
problem Quantifying how scalar curvature changes under C0 convergence in 3D. method Using harmonic functions and classical elliptic PDE estimates to show stability under C0 perturbations of the metric. result Explicitly quantifies the preservation of scalar curvature lower bounds under C0 convergence of metrics. 3D steady gradient Ricci solitons are all O(2)-symmetric.
problem Characterizing 3D steady gradient Ricci solitons.
method Analyzing asymptotic behavior and using O(2) symmetry.
result All 3D steady gradient Ricci solitons are O(2)-symmetric.
DreamFusion uses text-to-image diffusion models to create 3D images efficiently.
problem Lack of large-scale 3D datasets and efficient architectures for 3D synthesis.
method Adapting a 2D diffusion model to 3D synthesis using a loss based on probability density distillation.
result A 3D model can be optimized from a 2D diffusion model, allowing for text-to-3D synthesis.
3D Convolutional Neural Networks (3D-CNN) have been used for object recognition based on the voxelized shape of an object. However, interpreting the decision making process of these 3D-CNNs is still an infeasible task. In this paper, we present a unique 3D-CNN based Gradient-weighted Class Activation Mapping method (3D…
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…
Graph Neural Networks improve 3D object detection in LiDAR point clouds.
problem Challenges in processing LiDAR data due to its 3D geometry and massive volume.
method Proposes a Graph Neural Network (GNN) based framework for 3D object detection.
result GNNs successfully identify objects in 3D LiDAR point clouds.
The study connects knot complements to 3d theories via half-index calculations.
problem Understanding the relationship between knot complements and 3d theories.
method Using half-index calculations and inverted Habiro series, the study realizes knot complements as homological blocks.
result The colored Jones polynomial is derived from choosing specific poles in the half-index integral expression.
GCDM generates valid large 3D molecules and optimizes existing molecules.
problem Lack of geometric properties in 3D molecule generation models.
method Introduces Geometry-Complete Diffusion Model (GCDM) using equivariant GNNs.
result Significantly outperforms existing models in 3D molecule generation and optimization.