The paper generalizes Hotelling's theorem to harmonic manifolds and Damek-Ricci spaces.
problem The volume of tubes about curves in harmonic manifolds.
method Generalization of Hotelling's theorem to harmonic manifolds and computation of tube volumes in Damek-Ricci spaces.
result Harmonic manifolds have the tube property, and the volume of tubes can be computed.
Study on non-Gromov hyperbolic tube domains and their geometric properties.
problem Characterizing non-Gromov hyperbolic tube domains with convex bases.
method Provided a criterion for non-Gromov hyperbolicity, studied Hilbert metric, and continuity properties of complex geodesics.
result Similarity of geometry of tube domains and convex domains, connections between metrics.
Study on metric properties near boundary of tube domains.
problem Behavior of complete Kahler-Einstein metric near boundary.
method Estimates of metric and holomorphic bisectional curvatures.
result Obtained estimates near weakly pseudoconvex boundary points.
Classifies tube domains with specific properties in complex spaces.
problem Classifying tube domains with unique geometric properties.
method Analyzes tube domains in complex spaces with homogeneous boundaries and specific Levi forms.
result Classifies tube domains with large automorphism groups in arbitrary dimensions.
Characterizes harmonic manifolds using tube properties.
problem Understanding harmonic manifolds through tube properties.
method Analyzes properties of tubes in harmonic manifolds and geodesic segments.
result Characterizes harmonic manifolds using tube properties for geodesic segments and arc lengths.
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.
Study on tubes with specific Gauss map properties in 3D space.
problem Characterizing surfaces with a specific Gauss map property in 3D space.
method Analyzing tubes in Euclidean 3-space with Gauss map n satisfying ΔIn = Λn.
result Circular cylinders are the only surfaces of coordinate finite I-type Gauss map.
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…
The paper characterizes D'Atri spaces using total scalar curvature of hemispheres.
problem Characterizing D'Atri spaces using geometric properties.
method Characterization of D'Atri spaces via total scalar curvature of geodesic hemispheres.
result A 3D Riemannian manifold is a D'Atri space if and only if the total scalar curvature of tubes about geodesic segments holds.
Analyzes constant heat flow properties on Riemannian manifolds.
problem Classify manifolds with constant heat flow property.
method Analyzes overdetermined parabolic problems and uses isoparametric tubes.
result Analytic manifolds with constant flow property are isoparametric tubes.
Paper introduces a new SVR model using a combined reward and penalty loss function.
problem Regression problem, particularly handling data points outside and inside ε-tube.
method Combined reward cum penalty loss function to penalize and reward data points.
result Experimental results support the model's properties and effectiveness.
Weyl's tube formula holds for various cross-sections under symmetry conditions.
problem Can the volume of tubes around submanifolds be calculated for non-round cross-sections?
method Investigated the volume of tubes with general cross-sections D under symmetry conditions.
result The volume of tubes around submanifolds can be calculated for general cross-sections under symmetry conditions.
Computes tube formulas for valuations in complex space forms.
problem Computing values of valuations on complex space forms.
method Develops tube formulas for valuations in complex space forms and generalizes classical formulas.
result Generalizes classical formulas of Weyl, Gray and others.
The paper defines marginal tubes and proves their null nature.
problem Understanding the geometry of spacelike surfaces in spacetimes.
method Introducing marginal tubes and studying spacelike surfaces with double null coordinates.
result If every spacelike section of a marginal tube is a marginal surface, then the marginal tube is null.
Study preserves tube shape in symmetric spaces.
problem Preserving tube shape in symmetric spaces.
method Volume-preserving mean curvature flow.
result Tubeness preserved under certain conditions.
Study preserves tubeness of tubes in certain symmetric spaces.
problem Preserving the shape of tubes in symmetric spaces.
method Volume-preserving mean curvature flow, analyzing radius function and gradient.
result Tubeness is preserved in rank one symmetric spaces of non-compact type.
Study proves certain Zoll manifolds with entire Grauert tubes are isometric.
problem Characterizing Zoll manifolds with entire Grauert tubes.
method Computed algebraic indices of tangent bundles; proved isometry.
result Any Zoll manifold of type HP^2 with entire Grauert tube is isometric to canonical HP^2.
Zero entropy found in entire Grauert tubes of certain manifolds.
problem Analyzing the geodesic flow on Grauert tubes with zero entropy.
method Examined entire Grauert tubes on real analytic Riemannian manifolds.
result Geodesic flow on entire Grauert tubes has zero topological entropy.
Partial answer to affineness of entire Grauert tubes, with Stein manifold criterion.
problem Affineness of entire Grauert tubes
method Generalized Demailly's criterion for Stein manifolds
result Complement of a codimension-one subset is affine
Study on SU(2) Lie group tubes with left-invariant metrics.
problem Determine when SU(2) Grauert tubes are entire.
method Analyze left-invariant metrics on SU(2) with complete geodesic flow.
result Find a new obstruction to SU(2) tubes being entire.
Lower bound on boundary injectivity radius for specific tubes.
problem Estimating the boundary injectivity radius of Margulis tubes.
method Using curvature bounds to derive a lower bound.
result A lower bound on the boundary injectivity radius is provided.
New method for flexible tubes and structures, enabling rigid-foldability.
problem Creating flexible tubes with rigid-foldability.
method Discrete, semi-discrete, and smooth construction of surfaces (T-hedra and profile-affine surfaces).
result Unified treatment of continuous flexible structures composed of tubes.
Zoll manifolds with entire Grauert tubes are proven to be standard complex projective spaces.
problem Characterizing Zoll manifolds with entire Grauert tubes.
method Isometric comparison to CPn with the canonical metric. result Zoll manifolds of type CPn with entire Grauert tubes are isometric to CPn. Characterizes Calabi-Yau Hodge structures over tube domains.
problem Characterizing Hodge structures over specific domains.
method Using characteristic forms to characterize variations of Hodge structures.
result Characteristic forms uniquely identify Calabi-Yau Hodge structures over tube domains.
Study on volume of tubes and concentration in Riemannian geometry.
problem Understanding concentration loci in Riemannian manifolds and their relation to tube volumes.
method Provided a general formula for tube volumes, specialized to totally geodesic submanifolds, and investigated concentration loci.
result Explicitly proved concentration for codimension one cases and explored characterizations in Wasserstein and Box distances.
Sharp bounds found for distances between specific geometric shapes in hyperbolic space.
problem Finding effective distances between specific geometric shapes (tori) in hyperbolic 3-manifolds.
method Sharp, effective bounds on distances between tori of fixed injectivity radius.
result Effective bounds on distances between specific geometric shapes in hyperbolic space.
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…
Formula for tube volume on special geometric manifolds.
problem Calculating volumes of geometric structures.
method Using Einstein constant and polarization volume.
result Formula for renormalized tube volume.
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…
Let f:Mm⟶Rm+k be an immersion where M is a smooth connected m-dimensional manifold without boundary. Then we construct a subspace Ω(f) of Rk, namely push-out space. which corresponds to a set of embedded manifolds which are either parallel to f, tubes around f or, in…
We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold P of CPn which depends only on the radius of the tube, the degrees of the polynomials defining P and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the mo…
Extends tube volume estimates using integral curvature bounds.
problem Estimating volumes of tubes around submanifolds.
method Generalizes Heintze-Karcher inequality to k-Ricci curvature bounds. result Volume estimates for tubes around submanifolds using integral curvature bounds.
Let G be a complex Lie group, G_R a real form of G and X a G_R-stable domain of holomorphy in a complex G-manifold. If there is a G_R-invariant strictly plurisubharmonic function on X which has certain exhaustion properties, then we show that the extended domain G.X is also a domain of holomorphy. As an application we …
Estimates Betti numbers of loop spaces of compact manifolds.
problem Estimating Betti numbers of loop spaces of compact manifolds.
method Using finite Grauert tubes to provide an effective estimate.
result Implication of polynomial estimate in the limit of tube radius.
Researchers solved a geometry paradox for creased tubes.
problem Resolving the paradox of Gaussian curvature in creased tubes.
method Calculated Gaussian curvature in terms of rate of change of solid angle, dependent on fold angle and curvature.
result Gaussian curvature is zero overall despite the surface being doubly-curved.
In this paper, we study the spectrum of quantum tubes. Under certain intrinsic assumptions of the asymptotically flat submanifold of the Euclidean space, we prove the existence of the ground state of the quantum tube. The work is a generalization of Duclos, Exner and Krejcirik (CMP, 223(1), 13-28, 2001) and ourselves(m…
The paper classifies hypersurfaces with special curvature properties in various spaces.
problem Characterizing hypersurfaces with specific curvature properties.
method Introducing Ribaucour partial tubes and using them to derive applications.
result Classification of hypersurfaces of Enneper type in Rn+1. 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…
New formula for curvatures of curves in n-dimensional space.
problem Calculating curvatures of curves in high-dimensional spaces.
method Explicit formula for curvatures using derivatives.
result Generalization of Pappus' theorems to higher dimensions.
Explicit Taylor series for the volume of tubes in Lie groups
problem Computing the volume of tubes in riemannian manifolds
method Using bi-invariant metrics
result Explicit Taylor series for the volume of a tube in a Lie group
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…
Study constant mean curvature tubes in homogeneous spaces.
problem Global geometry of constant mean curvature tubes.
method Screw-motion invariants, foliation, numerical isoperimetric profile.
result Foliation result and embeddedness proof.
We establish the adiabatic dissapearance of Seiberg-Witten tunnelings on tubes R x N, where N is an S^1 fibration over a Riemann surface.
The study shows that the visible range from a point on harmonic manifolds follows an exponential distribution.
problem Understanding the visible range from a point on harmonic manifolds.
method Analyzing Poisson Boolean models on harmonic manifolds, focusing on the geometric mechanism of tube volumes around geodesic segments.
result The visible range from a point on harmonic manifolds follows an exponential distribution.
The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating with respect to the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincid…
We obtain various estimates of the life-time of two-dimensional minimal tubes in R^3 by potential theory methods.
The study examines constant mean curvature tubes around geodesics in specific 3-manifolds.
problem Investigating constant mean curvature surfaces in homogeneous 3-manifolds.
method Analyzing horizontal tubes foliating spaces under certain conditions.
result Horizontal tubes foliate spaces under specific curvature conditions.