arXiv research

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

We show that for a surface S, the subgraph of the pants graph determined by fixing a collection of curves that cut S into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made by Aramayona, Parlier, and Shackleton and has…

The paper studies spectral properties of Jacobi operator for surfaces with nonpositive Euler characteristic.

We develop a novel analogue of Euclidean PCA (principal component analysis) for data taking values on a Riemannian symmetric space, using totally geodesic submanifolds as approximating lower dimnsional submanifolds. We illustrate the technique on n-spheres, Grassmannians, n-tori and polyspheres.

We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Rado…

This research studies end-periodic mapping tori and their hyperbolic structures.

In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the inv…

Study on OI surfaces with unique geometric properties.

We construct smooth fiber bundles such that the fibers are exotic tori and the total space has finite abelian fundamental group. This gives examples of a Riemannian foliation on a closed manifold whose leaves are exotic tori and whose total space has finite abelian fundamental group.

The paper proves uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.

We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

In this paper we show that all totally real superconformal minimal tori in $CP^2$ correspond with doubly-periodic finite gap solutions of the Tzitzeica equation $ω_{z{\overline z}}=e^{-2ω}-e^ω$ Using the results on the Tzitzeica equation in integrable system theory, we describe explicitly all these tori by Prym-theta…

Proves geodesic connections on 2-torus without invariant tori.

Flat torus triangulations' space is homotopy equivalent to a torus.

Study minimal surfaces in 4D, find specific tori with total curvature -8π.

Extends Masur's divergence theorem to complex tori and Kummer surfaces.

We show that if a connected compact kählerian surface $M$ with nonpositive gaussian curvature is furnished with a closed conformal vector field $ξ$ whose singular points are isolated, then $M$ is isometric to a flat torus and $ξ$ is parallel. We also consider the case of a connected complete kählerian manifod $M$ of co…

In this paper, we generalize Medos-Wang's arguments and results on the mean curvature flow deformations of symplectomorphisms of $\CP^n$ in \cite{MeWa} to complex Grassmann manifold $G(n, n+m;\C)$ and compact totally geodesic Kähler-Einstein submanifolds of $G(n, 2n;\C)$ such as irreducible Hermitian symmetric spaces $…

We study homologically maximizing timelike geodesics in conformally flat tori. A causal geodesic $γ$ in such a torus is said to be homologically maximizing if one (hence every) lift of $γ$ to the universal cover is arclength maximizing. First we prove a compactness result for homologically maximizing timelike geodesics…

Given a closed Riemannian manifold (M, g), there is a partition Σ_i of its tangent bundle TM called the focal decomposition. The sets Σ_i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note,…

Study geodesics on flat tori, focusing on orthogonal lengths and their distribution.

Study on totally real flat minimal surfaces in quaternionic projective space.

Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its set of maximal flat totally geodesic tori, and $\mathrm{ad}(M)$ its adjoint space. We show that the kernel of the maximal flat Radon transform $τ:L^2(M) \rightarrow L^2(M')$ is precisely the orthogonal complement of the image of the pullback map…

Study on cylindrical handlebody-knots with symmetry and rigidity properties.

Study geodesics on flat tori, focusing on convex bodies.

We show the existence of special Lagrangian tori on one family of Borcea-Voisin threefolds. We also construct a family of special Lagrangian submanifolds on the total space of the canonical line bundle of projective spaces.

Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori. However, the identity was trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper we adapt t…

By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.

Researchers prove spectral rigidity of Liouville tori under specific conditions.

The study examines minimal surfaces in Riemannian products of surfaces.

Classifies totally geodesic submanifolds in Hopf-Berger spheres.

A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and inve…

We describe a new approach to the study of the set of all simple geodesics on a hyperbolic punctured torus. We introduce a valuation on the first integral homology group of the torus. This valuation associates to each homology class the length of the unique simple geodesic in it. We show that this valuation extends to …

We give a basic treatment of lattices $Γ$ in these groups. Certain tori $T_F$ and $T_B$ provide the model fiber and the base for a submersion of $Γ\backslash N$. This submersion may not be pseudoriemannian in the usual sense, because the tori may be degenerate. We then begin the study of periodic geodesics in these com…

Study of critical tori for mean curvature energies in Killing submersions.

Classifies totally geodesic submanifolds in symmetric spaces.

New example of manifolds with monotonic heat kernels found.

We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space.

Totally geodesic surfaces found in knots and links.

The study links Ricci curvature and convexity in complex tori.

Hedlund constructed Riemannian metrics on n-tori, $n \geq 3$ for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics are optimal for nilpotent fundamental groups.

Finite totally geodesic hypersurfaces in curved manifolds proven.

Classifies totally geodesic submanifolds in exceptional symmetric spaces.

Paper solves degenerated circle pattern metric problem in spherical geometry.

Classifies totally geodesic submanifolds in specific geometric spaces.

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. …

Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial result…

Consider a compact locally symmetric space $M$ of rank $r$, with fundamental group $Γ$. The von Neumann algebra $\vn(Γ)$ is the convolution algebra of functions $f\in\ell_2(Γ)$ which act by left convolution on $\ell_2(Γ)$. Let $T^r$ be a totally geodesic flat torus of dimension $r$ in $M$ and let $Γ_0\cong\bb Z^r$ be t…