arXiv research

Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.

We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for $δ$-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second f…

This paper classifies Kaehler submanifolds in hyperbolic space with low codimension.

Minimal Kaehler submanifolds in low codimension are often minimal.

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.

We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the la…

We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. W…

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation on a smooth vector bundle, i.e., one whose kernel consists only of ineffective arrows. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as a…

The paper classifies and studies conformal variations of submanifolds.

In Heisenberg groups, rectifiability is studied for subsets using $C^{1,α}$-regular surfaces.

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with cor…

We proved that a conformal immersion of $M_0^{n_0}\times M_1^{n_1}$ as an hipersurface in a Euclidean space must be an extrinsic product of immersions, under the assumption that $n_0, n_1 \geq 2$ and that $M^{n_0}_0\times M^{n_1}_1$ is not conformally flat. We also stated a similar theorem for an arbitrary number of fa…

Develops a spectral sequence for Lie group actions on manifolds.

A basic question in submanifold theory is whether a given isometric immersion $f\colon M^n\to\R^{n+p}$ of a Riemannian manifold of dimension $n\geq 3$ into Euclidean space with low codimension $p$ admits, locally or globally, a genuine infinitesimal bending. That is, if there exists a genuine smooth variation of $f$ by…

Confirming a conjecture, we show fundamental groups of certain abelian differentials are framed mapping class groups.

We study the second fundamental form of semi-isometric CR immersions from strictly pseudoconvex CR manifolds into Kähler manifolds. As an application, we give a precise condition for the CR umbilicality of real hypersurfaces, extending an well-known theorem by Webster on the nonexistence of CR umbilical points on gener…

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 statement clearly cannot be true for $C^2$ embeddings in general, due to the classi…

Derivative map for disk diffeomorphisms induces nontrivial homotopy groups.

The paper confirms a conjecture about submanifolds in Euclidean space.

The paper proves the behavior of the second fundamental form for Kaehler submanifolds in Euclidean space.

This thesis consists of two parts which share only a slight overlap. The first part is concerned with the study of ideals in the ring $C^\infty(M,R)$ of smooth functions on a compact smooth manifold M or more generally submodules of a finitely generated $C^\infty(M,R)$-module V. We define a topology on the space of all…

In this paper, we study degenerate CR embeddings $f$ of a strictly pseudoconvex hypersurface $M\subset \bC^{n+1}$ into a sphere $\bS$ in a higher dimensional complex space $\bC^{N+1}$. The degeneracy of the mapping $f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant deriv…

Optimizes bounds for threefold singularity volumes.

Study describes singularities of height functions on specific singular surfaces.

The paper extends affine connection results to singular warped and twisted products.

New singularity concept in GR: volume singularities.

The study shows stability of neckpinch singularities in mean curvature flows.

Study describes singularities of distance squared functions on singular surfaces.

In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on $\mathcal{SG}$ which induce the bijection onto e…

Proves positive mass theorem for AF spin manifolds with conical singularities.

The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.

Rectifies flat singular points of area-minimizing currents with singularity degree > 1.

Paper generalizes a theorem for real analytic singularities.

The paper studies singularities of pedal curves of hyperbolic frontals.

The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.

Maxfaces can have cuspidal edges near certain singularities.

New invariant distinguishes singular knots and links.

Study on singular twisted links and virtual braids, extending knot theory concepts.

Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.

Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…

We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

Classifies neighborhoods around specific leaf structures.

We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic $T_7$ singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical $A-D-E$ singularities of planar…

The article investigates conditions for isomorphism of singular tangent bundles.

Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.

This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…

Curvature criteria for A-simple singularities and their parallel curves identified.