Related papers: Embedding Some Riemann Surfaces into \C^2 with Int…
The question of adaptive mesh generation for approximation by splines has been studied for a number of years by various authors. The results have numerous applications in computational and discrete geometry, computer aided geometric design,…
To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the…
We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations…
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…
We prove the existence of a one parameter family of minimal embedded hypersurfaces in $R^{n+1}$, for $n \geq 3$, which generalize the well known 2 dimensional "Riemann minimal surfaces". The hypersurfaces we obtain are complete, embedded,…
We prove the second order differentiation formula along geodesics in finite-dimensional $RCD(K,N)$ spaces. Our approach strongly relies on the approximation of $W_2$-geodesics by entropic interpolations and, in order to implement this…
In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by $C^1$ isometric embeddings. This…
We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the H\"older space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $\alpha > \frac{1}{2}$ the…
The I-patch is a multi-sided surface representation, defined as a combination of implicit ribbon and bounding surfaces, whose pairwise intersections determine the natural boundaries of the patch. Our goal is to show how a collection of…
Certain semi-Riemannian metrics can be decomposed into a Riemannian part and an isochronal part. The properties of such metrics are particularly easy to visualize in a coordinate-free way, using isometric embedding. We present such an…
Given an $m$-dimensional closed connected Riemannian manifold $M$ smoothly isometrically immersed in an $n$-dimensional Riemannian manifold $N$, we estimate the diameter of $M$ in terms of its mean curvature field integral under some…
We are interested in the challenging problem of modelling densities on Riemannian manifolds with a known symmetry group using normalising flows. This has many potential applications in physical sciences such as molecular dynamics and…
In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on $\R^d$, such as radial basis functions (RBFs), to a…
With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral…
A systematic construction of higher order splines using two hierarchies of polynomials is presented. Explicit instructions on how to implement one of these hierarchies are given. The results are limited to interpolations on regular,…
A method is presented for forming polynomial interpolants on squares and cubes, which are more efficient in the so-called Euclidean degree than other commonly used methods with the same number of collocation points. These methods have…
Under consideration methods of constructing trigonometric interpolation splines of two variables on rectangular areas. These methods are easily generalized to the case of trigonometric interpolation splines of several variables on such…
In this paper, we propose a simple acceleration scheme for Riemannian gradient methods by extrapolating iterates on manifolds. We show when the iterates are generated from Riemannian gradient descent method, the accelerated scheme achieves…
For $X \sim X(n; 1, n^{-\alpha_1}, n^{-\alpha_2}, ...)$ in the multiparameter random simplicial complex model we establish necessary and sufficient strict inequalities on the $\alpha_i$'s to linearly embed the complex into…
We provide integral curvature bounds for compact Riemannian manifolds that allow isometric immersions into a Euclidean space with low codimension in terms of the Betti numbers.