arXiv research

The paper surveys partial results on convex real projective orbifolds with specific ends.

Study on totally geodesic planes in hyperbolic 3-manifolds with specific properties.

In this paper we prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description of minimal ends of finite total curvature in H^2xR. We also prove that a minimal …

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…

New construction of minimal surfaces in hyperbolic space.

A real projective orbifold is an $n$-dimensional orbifold modeled on $\mathbb{RP}^n$ with the group $PGL(n+1, \mathbb{R})$. We concentrate on an orbifold that contains a compact codimension $0$ submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed $(n-1)$-dimensional orbifolds times …

Classifies totally geodesic submanifolds in specific geometric spaces.

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…

Unified proof of end-point estimates for Radon transform on curved spaces.

Study projective deformations of hyperbolic 3-orbifolds with turnover ends.

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $SL(n+1, \mathbb{R})$ or $PGL(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The …

The paper proves a bound on the length of the shortest geodesic flower on certain manifolds.

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…

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…

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

Characterizes minimal surfaces with finite curvature in specific spaces.

Improved assessment of knee osteoarthritis using geodesic B-score.

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…

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

In this paper we develop the theory of properly immersed minimal surfaces in the quotient space $\mathbb H^2\times\mathbb R/G,$ where $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in $\mathbb H^2$ without fixed points. The horizontal isometry can be either a parabolic tra…

Classifies totally geodesic submanifolds in Hopf-Berger spheres.

We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness …

Classifies totally geodesic submanifolds in symmetric spaces.

Totally geodesic surfaces found in knots and links.

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.

Arithmetic 3-manifolds with infinite geodesics are proven.

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

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…

Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.

Totally geodesic Lagrangian submanifolds in nearly Kähler S³×S³.

Upper bound found for geodesic curvature on convex surfaces.

A metric Lie algebra g is a Lie algebra equipped with an inner product. A subalgebra h of a metric Lie algebra g is said to be totally geodesic if the Lie subgroup corresponding to h is a totally geodesic submanifold relative to the left-invariant Riemannian metric defined by the inner product, on the simply connected …

The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.

Study on geodesic surfaces in hyperbolic 3-orbifolds, proving overlaps in area sets.

One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds $M_+,M_-$ introduced and studied by Chen and Nagano in [Totally geodesic submanifolds of symm…

Classifies special submanifolds in geometric spaces.

Study counts geodesic surfaces in knot complements, finding unique ones for small knots.

Totally geodesic subvarieties in moduli spaces are studied.

Study on embedding hyperbolic 2-orbifolds in Bianchi orbifolds.

Classifies special hypersurfaces in Gödel spacetimes.

Study constructs totally geodesic submanifolds in skew position.

Classifies special hypersurfaces in specific types of manifolds.

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

Totally geodesic null hypersurfaces found in Lorentzian manifolds.

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Rieman…

We give a full description of totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold of constant curvature and present a new class of a cylinder-type totally geodesic submanifolds in the general case.