arXiv research

The aim of this note is to explain a generalization to the real case of a well known result on the automorphism group of an unbounded tube type symmetric domain in a complex vector space of finite dimension.

Classifies tube domains with specific properties in complex spaces.

We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between correspondin…

The study examines constant mean curvature tubes around geodesics in specific 3-manifolds.

Zoll manifolds with entire Grauert tubes are proven to be standard complex projective spaces.

Study on SU(2) Lie group tubes with left-invariant metrics.

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a rigidity result of Burger, Iozzi and Wienhard.

Study on tubes with specific Gauss map properties in 3D space.

Study proves certain Zoll manifolds with entire Grauert tubes are isometric.

Characterizes Calabi-Yau Hodge structures over tube domains.

Explicit Taylor series for the volume of tubes in Lie groups

In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermit…

New theory classifies knotted spheres in 4D space.

Maximal representations in exceptional Hermitian Lie groups classified for complex hyperbolic lattices.

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the …

We consider the Johnson-Koranyi-Hua system on symmetric Siegel domains of type two. We prove that all functions which are annihilated by the system and satisfy an H^2 integrability condition are pluriharmonic. So the situation is completely different on type two domains than on tube type domains: it was proved by Johns…

New method for flexible tubes and structures, enabling rigid-foldability.

We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Pruefer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs…

Maximal representations in infinite dimensional Hermitian spaces are studied with boundary maps.

Study on Gauss map surfaces in 3D space, focusing on anchor rings.

Zero entropy found in entire Grauert tubes of certain manifolds.

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. They are defined in terms of invariants in bounded cohomology and extend considerably the scope of maximal representations. We prove that weakly maximal representations are d…

This paper characterizes Kashiwara-Vergne groups using algebraic structures of knotted tubes.

This paper defines maximal measurable cocycles for surface groups into Hermitian Lie groups and studies their algebraic hulls.

Proves Steiner and tube formulae for 3D contact sub-Riemannian surfaces.

We report on new numerical computations of the set of self-contacts in tightly knotted tubes of uniform circular cross-section. Such contact sets have been obtained before for the trefoil and figure eight knots by simulated annealing -- we use constrained gradient-descent to provide new self-contact sets for those and …

This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.

Equivalence proven between uniformizing varieties and tensors, generalizing uniformization results.

The paper studies sub-Riemannian geometry and proves a Weyl's invariance result for Heisenberg groups.

Proves gap rigidity theorem for Hermitian symmetric spaces.

We define the Toledo invariant of a G-Higgs bundle on a Riemann surface, where G is a real semisimple group of Hermitian type, and we prove a Milnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley corre…

Generalizes classifying spaces for topological groups with torsion.

We develop the theory of maximal representations of the fundamental group of a compact connected oriented surface with boundary, into a group of Hermitian type. For any such representation we define the Toledo invariant, for which we establish properties such as uniform boundedness on the representation variety, additi…

A knot K in 1-bridge position with respect to a genus-g Heegaard surface in a 3-manifold can be moved by isotopy through knots in 1-bridge position until it lies in a union of n parallel genus-g surfaces tubed together by n-1 straight tubes, with K intersecting each tube in two arcs connecting the ends. We prove that t…

The paper classifies hypersurfaces with special curvature properties in various spaces.

Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-link…

The study shows that the visible range from a point on harmonic manifolds follows an exponential distribution.

Study shows only one type of proper domain in certain spaces.

The $(1,1)$-length of a knot is a braid group invariant equaling its level number.

The paper parametrizes spaces of maximal framed representations for a specific type of surface group.

The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.

Study of magnetic curves in SL(2,R) with quantization and horocycle projections.

We give necessary conditions for certain real analytic tube generic submanifolds in C^n to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in C^n. During the proof, we show that the local CR automorphism group of a minimal, finitely n…

Proof of Knot Entropy Conjecture for tube lattice polygons.

Let $Γ$ be a nondegenerate geodesic in a compact Riemannian manifold $M$. We prove the existence of a partial foliation of a neighbourhood of $Γ$ by CMC surfaces which are small perturbations of the geodesic tubes about $Γ$. There are gaps in this foliation, which correspond to a bifurcation phenomenon. Conversely, we …

We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisation…

Weyl's tube formula holds for various cross-sections under symmetry conditions.

We construct and identify star representations canonically associated with holonomy reducible simple symplectic symmetric spaces. This leads the a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley type and Hermitian symmetric spaces of tube type.