arXiv research

Study on embedding hyperbolic 2-orbifolds in Bianchi orbifolds.

In this note it is shown that every 7-dimensional Eschenburg space can be totally geodesically embedded into infinitely many topologically distinct 13-dimensional Bazaikin spaces. Furthermore, examples are given which show that, under the known construction, it is not always possible to totally geodesically embed a pos…

The study finds totally geodesic surfaces in hyperbolic 3-manifolds and verifies a conjecture.

Totally geodesically embeddings of infinitely many closed 7-manifolds into 13-dimensional positively curved closed Riemannian manifolds are constructed. The problems of computing pinching constants and existence of other totally geodesical embeddings are discussed.

Totally geodesic surfaces found in knots and links.

We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.

Classifies totally geodesic submanifolds in exceptional symmetric spaces.

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

New construction of minimal surfaces in hyperbolic space.

Our main theorem asserts that every Farey graph embedded in the 1-skeleton of the pants complex of any finite type surface is totally geodesic.

The paper proves the existence of capillary geodesics on Riemannian 2-disks.

Embeds Teichmüller space into geodesic currents, proving independence.

Holomorphic curves exiting bounded symmetric domains are asymptotically totally geodesic.

Geodesic surfaces embed into hyperbolic 3-manifolds for all finite group actions.

New Fuchsian groups found with special embedding properties.

The study proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.

We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic …

The study proves semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.

Study on embedding surfaces into 3-manifolds, focusing on equivariant cases.

Hyperbolic knots decompose into prism orbifolds.

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 study finds minimal volume hyperbolic 4-manifolds with embedded 3-manifolds.

Classifies hypersurfaces in Minkowski space with a specific foliation.

We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other nondegenerate summands. We also establish that every horizontal catenoid is nondegen…

In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S^3 for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get "as close as possible" to a counter-example. Specifically, we construct a sequence of hyperbolic knots {K_n} …

The paper embeds non-arithmetic hyperbolic manifolds into higher-dimensional spaces.

Critical nets in k-space have bounded edge lengths and vertices.

Study finds new minimal surfaces in Schwarzschild space.

We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces.

We construct a new Riemannian metric on Goldman space $\mathcal{B}(S)$, the space of the equivalence classes of convex projective structures on the surface $S$, and then prove the new metric, as well as the metric of Darvishzadeh and Goldman, restricts to be the Weil-Petersson metric on Teichm$\ddot{u}$ller space, embe…

We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic boundary, called the "turnover core,'' whose volume is bounded from above by a function…

Maximal representations link complex hyperbolic lattices to SU(p,q).

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's result concerning asymmetric complete minimal submanifolds of Euclidean space wi…

The study classifies real hypersurfaces in complex hyperbolic quadrics with isometric Reeb flow.

We prove the three embeddedness results as follows. $({\rm i})$ Let $Γ_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of $Γ_{2m+1}<2mπ$. In particular, the total curvature of $Γ_5<4π$ and thus any minimal surface $Σ\subset \…

Let $M$ be the image of a smooth CR embedding of a strictly pseudoconvex CR real hypersurface into a sphere. If the CR second fundamental form of $M$ vanishes, we show that $M$ is a totally geodesic submanifold.

In this paper, we prove the existence of maximal slices in anti-de Sitter spaces (ADS spaces) with small boundary data at spatial infinity. The main arguments is implicit function theorem. We also get a necessary and sufficient condition for boundary behavior of totally geodesic slice in ADS space. Moreover, we show th…

Develops a new theory of width for embedded circles in Riemannian manifolds.

We use pinched smooth hyperbolization to show that every closed, nonpositively curved $n$-dimensional manifold $M$ can be embedded as a totally geodesic submanifold of a closed, nonpositively curved $(n+1)$-dimensional manifold $\hat{M}$ of geometric rank one.

The paper proves the existence of non-trivial lamination in complex projective space.

Totally geodesic submanifolds cover only finitely many varieties in Teichmüller space.

Isometric embeddings of Teichmüller spaces are derived from branched coverings.

Classifies totally geodesic submanifolds in Hopf-Berger spheres.

Classifies totally geodesic submanifolds in symmetric spaces.

This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the Kähler class, is used to define and characterize a special class of totally geodesic embeddings, called "tight embeddings". In addition, special is…

New technique identifies submanifolds in symmetric spaces based on Ricci curvature.

We classify the polycyclic totally ordered simple dimension groups, i.e. dimension groups given by a dense embedding of n-dimensional lattice into the real line. Our method is based on the geometry of simple geodesics on the hyperbolic surface of genus greater or equal two. The main theorem says that isomorphism classe…

We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal uppe…