arXiv research

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

Zero entropy found in entire Grauert tubes of certain manifolds.

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

Computes tube formulas for valuations in complex space forms.

Partial answer to affineness of entire Grauert tubes, with Stein manifold criterion.

H. Hotelling proved that in the n-dimensional Euclidean or spherical space, the volume of a tube of small radius about a curve depends only on the length of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's theorem to rank one symmetric spaces computing the volumes of the tubes explicitly in these …

Estimates Betti numbers of loop spaces of compact manifolds.

Tubes in manifolds require wide spaces.

Study on volume of tubes and concentration in Riemannian geometry.

Researchers prove a complex geometric conjecture about certain manifolds.

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

To every real analytic Riemannian manifold M there is associated a complex structure on a neighborhood of the zero section in the real tangent bundle of M. This structure can be uniquely specified in several ways, and is referred to as a Grauert tube. We say that a Grauert tube is entire if the complex structure can be…

A formula of the renormalized volume of tubes over polalized Kähler-Einstein manifolds is given in terms of the Einstein constant and the volume of the polarization.

The study proves a tube theorem for complex hyperbolic manifolds.

We give sharp, effective bounds on the distance between tori of fixed injectivity radius inside a Margulis tube in a hyperbolic 3-manifold.

The paper characterizes D'Atri spaces using total scalar curvature of hemispheres.

Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of TM. We call such manifolds Grauert t…

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

Study Kähler manifolds on tube domains, proving curvature uniqueness and applications to optimal transport.

The authors showed in a preceding paper that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed on…

We correct and complete a conjecture of D. Gabai, R. Meyerhoff and N. Thurston on the classification and properties of thin tubed closed hyperbolic 3-manifolds. We additionally show that if N is a closed hyperbolic 3-manifold, then either N=Vol3 or N contains a closed geodesic that is the core of an embedded tube of ra…

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

We prove that elliptic tubes over properly convex domains of the real projective space are C-convex and complete Kobayashi-hyperbolic. We also study a natural construction of complexification of convex real projective manifolds.

The Weyl tube theorem is extended to Kähler manifolds.

We obtain a locally symmetric Kaehler Einstein structure on a tube in the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained Kaehler Einstein structure cannot have constant holomorphic sectional curvature.

Explicit Taylor series for the volume of tubes in Lie groups

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…

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

In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where "holonomy invariant" means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is consta…

The paper defines marginal tubes and proves their null nature.

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…

If a closed, orientable hyperbolic 3--manifold M has volume at most 1.22 then H_1(M;Z_p) has dimension at most 2 for every prime p not 2 or 7, and H_1(M;Z_2) and H_1(M;Z_7) have dimension at most 3. The proof combines several deep results about hyperbolic 3--manifolds. The strategy is to compare the volume of a tube ab…

Lower bound on boundary injectivity radius for specific tubes.

A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can ar…

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

The systole length of hyperbolic n-manifolds is bounded by a function of n and t.

In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the Turaev-Viro-Ocneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjuga…

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 …

Study equivariant 4-genus of knots in symmetric 4-manifolds.

We prove a conjecture of Menasco and Zhang that if a tangle is completely tubing compressible then it consists of at most two families of parallel strands. This is related to problems of graphs in 3-manifold. A 1-vertex graph $Γ$ in a 3-manifold $M$ with a genus 1 Heegaard splitting is standard if it consists of one or…

All knots with unknotting number ≤ 21 are smoothly slice in K3 surface.

Two codimension-one submanifolds are cobordant if they have the same homology class.

A surface is called a tube if its level-sets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the so-called flow vector. We study how the geometry of the Gaussian image of a higher-dimensional minimal tube M is controlled by the angle alph…

We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry …

Let $(M,ω)$ be a pseudo-Hermitian space of real dimension $2n+1$, that is $\RManBase$ is a $\CR-$manifold of dimension $2n+1$ and $ω$ is a contact form on $M$ giving the Levi distribution $HT(M)\subset TM$. Let $M^ω\subset T^*M$ be the canonical symplectization of $(M,ω)$ and $M$ be identified with the zero section of …

In this paper, we investigate the volume-prserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space. We prove that the tubeness is preserved along the flow under certain conditions.

We consider an example of tubes of hypersurfaces in Euclidean space and generalise the tube formula to supercase. By this we assign to a point of the hypersurface in superspace a rational characteristic function. Does this rational function appear when we calculate the zeta-function of an arithmetic variety?

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…