arXiv research

Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.

The paper explores isometric models and Busemann functions for Funk and Hilbert discs.

Isometric embeddings of Teichmüller spaces are derived from branched coverings.

Proof that specific groups are quasi-isometrically rigid.

Paper finds isometric timelike minimal surfaces with unique properties.

The paper explores rigidity and flexibility of isometric extensions with critical Hölder exponent.

Paper classifies pillow box isometric deformations preserving crease patterns.

We prove an analogue of Farb-Masur's theorem that the length-spectra metric on moduli space is "almost isometric" to a simple model $\mathcal {V}(S)$ which is induced by the cone metric over the complex of curves. As an application, we know that the Teichmüller metric and the length-spectra metric are "almost isometric…

In this paper we consider the Cauchy problem for isometric immersions. More precisely, given a smooth isometric immersion of a codimension one submanifold we construct $C^{1,α}$ isometric extensions for any $α<\frac{1}{n(n+1)+1}$ via the method of convex integration.

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity.

Study on isometric submanifolds with preserved Gauss map metrics.

Heat kernels map RCD spaces to Riemannian manifolds.

We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.

In trying to generalize Bianchi's Bäcklund transformation of quadrics to Bäcklund transformations of isometric deformations of other (classes of) surfaces, we investigate basic features of the isometric deformation of surfaces via the Bäcklund transformation with isometric correspondence of leaves of a general nature (…

The paper improves the approximation of isometric immersions in high codimension.

The paper examines isometric timelike surfaces in 4D Minkowski space.

Explains isometric immersions and their applications.

The paper examines rigidity of metric constructions in Wasserstein spaces.

New findings on minimal isometric immersions of flat n-tori into spheres.

Maps between Hadamard manifolds are quasi-isometric to harmonic maps.

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thu…

Revisits Weyl's problem on isometric immersions of spheres into 3D manifolds.

Study holomorphic isometric embeddings of a Grassmannian into quadrics.

Framework for isometric immersions of planar regions from framed curves.

Stability result for nearly isometric subspaces and Finsler surfaces.

The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…

Characterizes quasi-isometric embeddings in coarsely Lipschitz category.

In this article, we study holomorphic isometric embeddings between bounded symmetric domains. In particular, we show the total geodesy of any holomorphic isometric embedding between reducible bounded symmetric domains with the same rank.

We investigate the structure of real hypersurfaces with isometric Reeb flow in Kaehler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type.

Study minimal Kähler submanifolds in product of space forms.

Long spacelike embeddings can be approximated by isometric ones.

Proposes IIKL for preserving geometric properties of non-Euclidean data.

We study isometric Lie group actions on the compact exceptional groups E6, E7, E8, F4 and G2 endowed with a biinvariant metric. We classify polar actions on these groups. We determine all isometric actions of cohomogeneity less than three on E6, E7, F4 and all isometric actions of cohomogeneity less than 20 on E8. More…

We present some results on reductions and the copolarity of isometric group actions, which we obtained in our thesis. We also describe a resolution construction for isometric actions with respect to a reduction and give examples.

Research shows arc complex is not quasi-isometric to sphere complex.

To analyze high-dimensional and complex data in the real world, deep generative models, such as variational autoencoder (VAE) embed data in a low-dimensional space (latent space) and learn a probabilistic model in the latent space. However, they struggle to accurately reproduce the probability distribution function (PD…

First explicit isometric immersion of a flat Klein bottle in 3D space.

The paper extends isometric embedding results to null cones and spheres.

The paper explores almost paracomplex structures on 4-manifolds and their properties.

The paper proves isometric embeddings for smooth manifolds.

Study isometric immersions in 3D Lie groups, proving new characterizations and classifications.

We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.

Paper proves existence of isometric immersions for negatively curved surfaces with unbounded second fundamental form.

New criteria for non-isometric group actions in metric spaces.

Study on embedding properties of Riemannian manifolds with specific geometric constraints.

Let $Σ$ be a codimension one submanifold of an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We give a necessary condition for an isometric immersion of $Σ$ into $\mathbb R^q$ equipped with the standard Euclidean metric, $q\geqslant n+1$, to be locally isometrically $C^1$-extendable to $M$. Even if this cond…

We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the Hölder space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $α> \frac{1}{2}$ the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, wh…

Proves isometric embeddings in Euclidean spaces for RCD spaces.