Arithmetic 3-manifolds with infinite geodesics are proven.
problem Arithmeticity of 3-manifolds with infinitely many geodesics.
method Analysis of totally geodesic surfaces in hyperbolic 3-manifolds.
result Arithmetic 3-manifolds with infinite geodesics are proven.
Classifies totally geodesic submanifolds in symmetric spaces.
problem Classifying submanifolds in symmetric spaces.
method Classification of totally geodesic submanifolds in products of rank one symmetric spaces.
result Infinitely many examples of irreducible totally geodesic submanifolds in Hermitian symmetric spaces.
Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
The study finds infinitely many twist knot complements with totally geodesic surfaces.
problem Finding infinitely many twist knot complements with a specific number of totally geodesic surfaces.
method Using a family of twist knot complements and their dihedral covers, the authors construct examples of hyperbolic 3-manifolds with totally geodesic surfaces.
result The construction of infinitely many non-commensurable hyperbolic 3-manifolds with exactly k totally geodesic surfaces for any positive integer k.
Finite totally geodesic hypersurfaces in curved manifolds proven.
problem Characterizing totally geodesic hypersurfaces in curved manifolds.
method Analytic Riemannian manifold analysis with negative sectional curvature.
result Closed manifolds with negative curvature have only finitely many totally geodesic hypersurfaces.
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
We parametrize the commensurability classes of curves on Shimura surfaces that are totally geodesic, i.e., the commensurability classes of so-called C-Fuchsian subgroups. In particular, if a Shimura surface contains one commensurability class of totally geodesic curves, it contains infinitely many.
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…
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.
The paper solves curvature problems on infinite hyperbolic surfaces.
problem Prescribed curvature flow on hyperbolic surfaces with infinite topological type.
method Introduced a prescribed curvature flow adapted to infinite cellular decompositions.
result Established well-posedness of the flow and proved convergence under certain conditions.
The study examines surfaces in hyperbolic 3-manifolds that become nearly flat.
problem Characterizing surfaces in hyperbolic 3-manifolds that become nearly flat.
method Analyzes asymptotically geodesic surfaces in hyperbolic 3-manifolds with finite and infinite volume.
result For finite volume, asymptotically geodesic surfaces are dense; for infinite volume, they do not exist.
We study totally geodesic planes in hyperbolic 3-manifolds M having incompressible core and degenerate ends. We prove a Ratner-type phenomenon: a closed minimal PSL(2,R)−invariant subset of M is either an immersed totally geodesic surface or all of M. We also show that for an arbitrary infinite volume hyperboli…
Totally geodesic submanifolds in product spaces imply special curvature properties.
problem Characterizing manifolds based on the existence of totally geodesic submanifolds.
method Analyzing totally geodesic submanifolds in products of non-positively curved manifolds with transversality conditions.
result If infinitely or just a single dense such submanifolds exist, the manifold must be locally symmetric.
Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.
problem Geodesic preservation in infinite-dimensional manifolds.
method Definition of spray-invariant sets and analysis of their properties.
result Different geometric properties of spray-invariant sets based on their regularity.
Infinite hyperbolic manifolds share same perimeter-to-volume ratio.
problem Finding hyperbolic manifolds with a fixed perimeter-to-volume ratio.
method Constructing infinitely many hyperbolic manifolds with nonempty boundaries.
result Existence of incommensurable hyperbolic manifolds with a fixed perimeter-to-volume ratio.
We develop a Malliavin calculus on the horizontal path space of a totally geodesic Riemannian foliation. As a first application, under suitable assumptions, we prove a log-Sobolev inequality for a natural one-parameter family of infinite-dimensional Ornstein-Uhlenbeck type operators. As a second application, we obtain …
The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.
problem Studying spectral properties of the horizontal Laplacian
method Interpreting the horizontal Laplacian as a twisted Laplacian acting on a flat vector bundle
result The horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold
Arithmeticity proven for certain lattices in SO(n,1) with specific geometric properties.
problem Arithmeticity of lattices in SO(n,1) with totally geodesic subspaces.
method Superrigidity theorem for certain representations of lattices, using equidistribution results from homogeneous dynamics.
result Arithmeticity of lattices proven under specific geometric conditions.
Complex hyperbolic manifolds with many totally geodesic submanifolds are arithmetic.
problem Characterizing arithmeticity of complex hyperbolic manifolds with certain submanifolds.
method Developing superrigidity theorems for complex hyperbolic lattices and proving nonexistence of certain maps.
result Finite volume complex hyperbolic n-manifolds containing infinitely many maximal totally geodesic submanifolds of dimension at least two are arithmetic. Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.
problem Analyzing free boundary minimal hypersurfaces in the Riemannian Schwarzschild space.
method Variational methods and geometric analysis.
result Zero Morse index for certain free boundary rotationally symmetric totally geodesic hypersurfaces in the Riemannian Schwarzschild space.
The study finds infinite commensurability classes of hyperbolic manifolds with Salem number lengths.
problem Understanding commensurability classes of arithmetic hyperbolic manifolds.
method Analyzing Salem numbers and their relation to arithmetic hyperbolic manifolds.
result Infinitely many commensurability classes of arithmetic hyperbolic manifolds with geodesic lengths equal to logarithms of Salem numbers.
The paper explores subspaces in hyperbolic lattices and their arithmetic properties.
problem Arithmeticity criterion for hyperbolic lattices and suborbifolds.
method Analysis of totally geodesic suborbifolds and Vinberg's commensurability invariants.
result Arithmeticity of hyperbolic orbifolds is linked to the existence of infinitely many fc-subspaces.
Study infinite combinatorial Ricci flow on spherical surfaces.
problem Investigate infinite combinatorial Ricci flow with spherical background.
method Establish existence and convergence of solution for infinite cellular decompositions.
result Existence and convergence of solution for infinite combinatorial Ricci flow in spherical geometry.
This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of Xa,b=(H2)a×(H3)b. A special case describes all Shimura subvarieties of type A1 Shimura varieties. We produce, for any $n\geq 1…
Periodic geodesics on Hilbert half-Lie groups exist whenever the fundamental group is nontrivial.
problem Existence of periodic geodesics on Hilbert half-Lie groups
method Using completeness results and Lyusternik-Fet type theorem
result Periodic geodesics exist whenever the fundamental group is nontrivial
We construct the first examples of complete, properly embedded minimal surfaces in H2×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…
The paper studies geodesic hypersurfaces in hyperbolic manifolds and their fundamental groups.
problem Understanding the fundamental groups of hyperbolic manifolds through geodesic hypersurfaces.
method Analyzing sequences of asymptotically geodesic hypersurfaces and their properties.
result If a closed hyperbolic manifold contains a sequence of asymptotically geodesic hypersurfaces, its fundamental group is virtually special and linear over integers.
S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its pert…
In this article we investigate a first order reparametrization-invariant Sobolev metric on the space of immersed curves. Motivated by applications in shape analysis where discretizations of this infinite-dimensional space are needed, we extend this metric to the space of Lipschitz curves, establish the wellposedness of…
Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
Let (M,J) be an almost complex manifold. We show that the infinite-dimensional space Tau of totally real submanifolds in M carries a natural connection. This induces a canonical notion of geodesics in Tau and a corresponding definition of when a functional, defined on Tau, is convex. Geodesics in Tau can be expressed i…
Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.
problem Rigidity of geodesic planes in Hitchin manifolds.
method Constructing a specific surface group and analyzing its action on the Hitchin manifold.
result Existence of floating geodesic planes in Hitchin manifolds with fractal closures.
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.
Study on OI surfaces with unique geometric properties.
problem Characterizing and classifying ortho-integral surfaces.
method Analyzing geodesic arcs and cosh-length properties.
result Infinitely many commensurability classes of OI surfaces arise as topologies vary.
Classifies totally geodesic submanifolds in exceptional symmetric spaces.
problem Classifying totally geodesic submanifolds in exceptional symmetric spaces.
method Classification and introduction of an invariant (Dynkin index) for totally geodesic embeddings.
result Existence of a totally geodesic submanifold of minimal codimension with specific properties.
Paper solves degenerated circle pattern metric problem in spherical geometry.
problem Existence and rigidity of (degenerated) circle pattern metrics with prescribed total geodesic curvatures.
method Defined prescribed combinatorial Ricci flows and studied their convergence.
result First degenerated result for total geodesic curvatures in spherical background geometry.
In this paper, we introduce the associated geodesic-Einstein flow for a relatively ample line bundle L over the total space X of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair (X,L) is nonlinear semistable if the {associated} Donaldson …
Classifies totally geodesic submanifolds in specific geometric spaces.
problem Identifying totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
method Developed new techniques for studying totally geodesic submanifolds in analytic Riemannian manifolds, homogeneous spaces, and Riemannian cones.
result Obtained a classification of totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
The study finds totally geodesic surfaces in hyperbolic 3-manifolds and verifies a conjecture.
problem Finding totally geodesic surfaces in hyperbolic 3-manifolds.
method Developed algorithms to determine and verify the existence of totally geodesic surfaces.
result Discovered nine 3-manifolds with totally geodesic surfaces and verified Menasco-Reid's conjecture for knots up to 12 crossings.
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.
problem Characterizing minimal hypersurfaces in S^5 with certain curvature conditions.
method Analyzing hypersurfaces with constant scalar curvature and zero Gauss curvature.
result Minimal hypersurfaces in S^5 with these curvature properties are totally geodesic.
Totally geodesic Lagrangian submanifolds in nearly Kähler S³×S³.
problem Characterizing Lagrangian submanifolds in nearly Kähler manifolds.
method Analyzing H-umbilical properties and their implications for geodesicity. result In nearly Kähler S³×S³, H-umbilical Lagrangian submanifolds are totally geodesic. 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.
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 study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
The study finds infinitely many Shimura subvarieties in Jacobian loci for curves of genus 2, 3, and 4.
problem Understanding Shimura subvarieties in Jacobian loci for curves of positive genus.
method Analyzing Galois covers of curves and their Shimura subvarieties under specific numerical conditions.
result The Jacobian locus contains infinitely many Shimura subvarieties of positive dimension for g≤4. Study uses Zilber-Pink conjecture and dynamical methods to solve rigidity problems.
problem Rigidity problems for arithmetic hyperbolic lattices.
method Zilber-Pink conjecture and dynamical methods.
result New results about reconstructing Hodge structures from their loci.
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…