arXiv research

A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked if and only if it is not outer-planar. A graph is outer-flat if it can be embed…

Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.

We equip the whole tangent space $TM$ to a hyperbolic manifold $M$ (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of $M$ extend to isometries of $TM$ by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In t…

This paper proves energy convexity for bi-harmonic maps into spheres, with applications to heat flow and uniqueness.

We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^{n}$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as view…

Each compact manifold M of finite dimension k is differentiable and supports an intrinsic probability measure. There then exists a measurable transformation of M to the k-dimensional "surface" of the (k+1)-dimensional ball.

The paper proves height estimates for Killing graphs on manifolds.

Quantitative analysis of soccer players' passing ability focuses on descriptive statistics without considering the players' real contribution to the passing and ball possession strategy of their team. Which player is able to help the build-up of an attack, or to maintain the possession of the ball? We introduce a novel…

Reformulates Wasserstein autoencoder for hyperbolic latent space.

Many nonparametric regressors were recently shown to converge at rates that depend only on the intrinsic dimension of data. These regressors thus escape the curse of dimension when high-dimensional data has low intrinsic dimension (e.g. a manifold). We show that k-NN regression is also adaptive to intrinsic dimension. …

Proves rigidity of geodesic balls in spheres under certain deformations.

Active subspaces on Riemannian manifolds generalize Euclidean principles.

AdaRL improves robust RL by adaptively adjusting policy complexity.

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the…

The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.

Robotic arm learns to manipulate a ball by choosing goals from learned experience.

Develops a framework for generating harmonic maps from a unit ball to a sphere.

Sharp Veronese rigidity theorem for submanifolds of unit ball.

The paper confirms a conjecture about convex bodies and their properties.

Extended Tetrahedral Property to non-Euclidean spaces.

Study shows horofunction compactification's topology matches dual norm's unit ball.

Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.

Weyl's tube formula holds for various cross-sections under symmetry conditions.

Study semicontinuity of capacity in non-smooth spaces using intrinsic flat convergence.

New IPL graphs identified and conditions for their projective embeddings established.

We develop a novel Gaussian process method for manifold data.

Maps preserving mass and injective on boundary are isometries.

The paper examines vertical curves and fibers in the Heisenberg group, proving properties and constructing counterexamples.

New metrics assess class overlap and imbalance in datasets.

Sobolev GAN uses a new IPM to improve GAN performance in semi-supervised learning.

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the…

Let $N\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit ${\cal O}(p) =\{u p u^*: u\in U_M\}$ of the Jones …

We analyze the problem of sequential probability assignment for binary outcomes with side information and logarithmic loss, where regret---or, redundancy---is measured with respect to a (possibly infinite) class of experts. We provide upper and lower bounds for minimax regret in terms of sequential complexities of the …

The study shows properties of stable anisotropic minimal hypersurfaces in 4D space.

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved …

The paper extends manifold learning to arbitrary norms, improving molecular motion mapping.

Maps from spheres and disks to convex shapes via curvature flow.

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, whil…

We obtain area growth estimates for constant mean curvature graphs in $\mathbb{E}(κ,τ)$-spaces with $κ\leq 0$, by finding sharp upper bounds for the volume of geodesic balls in $\mathbb{E}(κ,τ)$. We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in $\mathbb{E}(κ…

3-balls in 4-sphere become isotopic in 5-ball.

We develop methods to efficiently approximate data in metric spaces without additional assumptions.

Study on ball widths and minimal submanifolds in space forms.

The paper compares isoperimetric quotients and capacities in weighted manifolds.

This paper describes a method to construct standard 4-balls from homotopy 4-balls in $\mathbb{C}^2$.

Diameters of ball intersections decrease as centers move apart.

Two minimal hypersurfaces in a ball intersect in any half-ball.

Many solutions found for a ball boundary problem.

New proofs show how to embed certain 3D shapes into a 4D ball.