arXiv research

The paper proves that certain geodesics pass through timelike poles on specific Lorentzian 2-tori.

Timelike geodesics foliate a specific Lorentzian 2-torus.

New Poisson manifolds created over 2-tori.

Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.

We use surgery along 2-tori embedded in a union of two copies of a product of punctured 2-tori to produce a new collection of homotopy 4-spheres (4-manifolds homotopy equivalent to $S^4$ and hence homeomorphic to $S^4$ but possibly not diffeomorphic to $S^4$). It is still unknown if these new examples are in fact exoti…

Minimal 2-tori in 4-sphere have at least index 6.

The study proves biharmonic unit sections on 2-tori are always harmonic and exists in each homotopy class.

The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies …

We prove existence results that give information about the space of minimal immersions of 2-tori into $S ^ 3$. More specifically, we show that \begin{enumerate} \item For every positive integer $n$, there are countably many real $n$-dimensional families of minimally immersed 2-tori in $S ^ 3$. Every linearly ful…

New optimal isosystolic inequality found for Finsler reversible 2-tori.

The study provides a structure theorem for a new class of noncompact 3-manifolds.

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Koda…

Improved bounds on $\Z_2$-torus actions on positively curved manifolds.

Study shows certain 4D hyperbolic links don't contain geodesic 3-manifolds.

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants, which are invariants of the topology of the manifold, of th…

We give new examples of closed smooth 4-manifolds which support singular metrics of nonpositive curvature, but no smooth ones, thereby answering affirmatively a question of Gromov. The obstruction comes from patterns of incompressible 2-tori sufficiently complicated to force branching of geodesics for nonpositively cur…

Tight embeddings of 2-tori in 3D space contain short loops.

This article has two purposes. The first is to give an expository account of the integrable systems approach to harmonic maps from surfaces to Lie groups and symmetric spaces, focusing on spectral curves for harmonic 2-tori. The most unwieldy aspect of the spectral curve description is the periodicity conditions and th…

We show that for every non-negative integer n there is a real n-dimensional family of minimal Lagrangian tori in CP^2, and hence of special Lagrangian cones in C^3 whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curv…

Study of diffeomorphisms groups on lens spaces with Morse-Bott foliations.

Manufacturing infinite sets of knotted and unknotted surfaces in 4-manifolds.

In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…

We discuss the minimum of Willmore functional of torus in a Riemannian manifold $N$, especially for the case that $N$ is a product manifold. We show that when $N=S^2\times S^1$, the minimum of $W(T^2)$ is 0, and when $N=R^2\times S^1$, there exists no torus having least Willmore functional. When $N=H^2(-c)\times S^1$, …

Defines metrics for Lorentzian spaces and explores maximal developments.

Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the classification of two-symmetric Lorentzian manifolds is explained; conformally …

Study gluing of Lorentzian length spaces and their causal ladder properties.

Defines doubling conditions for Lorentzian spaces linked to curvature bounds.

Research explores Lorentzian distances on a specific geometric plane.

The study examines conditions for high-dimensional Lorentzian cobordisms with specific structure groups.

The study proves a transverse diameter theorem for Lorentzian foliations.

Finds a limit on Lorentzian manifolds.

The paper classifies 3D Lorentzian $Ein(2)$ Lie groups.

Timelike minimal surfaces in Lorentzian Heisenberg group have singular points.

Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.

Defines new metrics for Lorentzian spaces and their convergence.

Compact group found for special Lorentzian manifolds.

Study on Lorentzian spaces with curvature bounds, proving comparison theorems.

Characterizes Lorentzian manifolds embeddable in Minkowski spacetime.

Diagonalizes metrics of 3D Lorentzian manifolds.

This note (originally from 2015) provides a proof of a 1985 conjecture of Montiel and Ros concerning the conformal volume of tori. This updated version adds a proof of the claim made in Remark 5 about the value of the conformal volume of tori in the cases not covered by the conjecture of Montiel and Ros. Originally, I …

We prove upper and lower bounds for the eigenvalues of the Dirac operator and the Laplace operator on 2-dimensional tori. In particluar we give a lower bound for the first eigenvalue of the Dirac operator for non-trivial spin structures. It is the only explicit estimate for eigenvalues of the Dirac operator known so fa…

Proves a synthetic Lorentzian Cartan-Hadamard theorem.

In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We in…

If an $m+2$-manifold $M$ is locally modeled on $\RR^{m+2}$ with coordinate changes lying in the subgroup $G=\RR^{m+2}\rtimes ({\rO}(m+1,1)\times \RR^+)$ of the affine group ${\rA}(m+2)$, then $M$ is said to be a \emph{Lorentzian similarity manifold}. A Lorentzian similarity manifold is also a conformally flat Lorentzia…

Proves existence of longest paths in sub-Lorentzian problems.

Study classifies Riemann solitons on specific 3D Lorentzian groups.