New sub-Riemannian spaces with boundary meet curvature-dimension condition.
problem Finding sub-Riemannian manifolds with boundary satisfying curvature-dimension condition.
method Constructing specific sub-Riemannian structures on half-spaces and hemispheres.
result Provided new examples of sub-Riemannian manifolds with boundary that meet RCD(K,N) condition. Counterexample disproves gluing theorem for MCP metric spaces.
problem Gluing theorems for MCP metric measure spaces are not universally valid.
method Used Grushin half-plane as a counterexample.
result The doubling of Grushin half-plane does not satisfy MCP(0,N) for all N.
We employ Grushin jets which are adapted to the geometry of Grushin-type spaces to obtain the existence-uniqueness of viscosity solutions to the ∞(x)-Laplace equation in Grushin-type spaces. Due to the differences between Euclidean jets and Grushin jets, the Euclidean method of proof is not valid in this environ…
Computed distortion coefficients for the α-Grushin plane.
problem Analyzing the distortion coefficients of the α-Grushin plane.
method Using generalised trigonometric functions and synthetic curvature conditions.
result Estimates for distortion coefficients and a curvature condition conjecture.
Researchers create metrics on spheres with Ricci curvature ≥1, limiting to Grushin hemisphere.
problem Constructing metrics with Ricci curvature ≥1 on spheres.
method Sequence of Riemannian metrics on Sm+n with Ric≥1. result Gromov-Hausdorff limit of the sequence is the Grushin hemisphere.
Study on quantum particle evolution on Grushin cylinder, embedding in R^3.
problem Understanding quantum particle behavior on Grushin cylinder.
method Intrinsic and extrinsic quantizations, embedding in R^3.
result Formulas to embed Grushin cylinder in R^3, useful for other purposes.
New cylindrical solutions found for Grushin-type problem.
problem Critical Grushin-type problem on CR sphere.
method Local Pohozaev identities for non-degeneracy, Lyapunov-Schmidt reduction for solutions.
result New type of multi-bubbling cylindrical solutions constructed.
Research finds double bubbles in Grushin plane with specific geometric properties.
problem Addressing double bubble problem in Grushin plane with anisotropic perimeter.
method Existence via direct method and characterization via first variation techniques.
result Angles at intersections satisfy 120-degree rule, with differences for α=0 and α=1.
The study examines optimal synthesis in a radially symmetric Grushin space with conditions on the weight function.
problem Optimal synthesis in a radially symmetric Grushin space with a weight function.
method Analysis of the geometry of R3 with a weighted Carnot-Carathéodory metric, providing conditions for Grushin-like structure, and describing optimal synthesis. result Sufficient conditions on the weight function ensure a Grushin-like structure, and the candidate cut time coincides with the true cut time in the integrable case.
Fundamental solutions found for p-Laplace equations in Heisenberg and Grushin spaces.
problem Finding solutions to p-Laplace equations with drift terms in specific geometric spaces.
method Analyzing fundamental solutions in the Heisenberg group and Grushin-type planes.
result Natural generalizations of Beals, Gaveau, and Greiner's solutions for the Laplace equation with drift term.
We realise the first and second Grushin distributions as symmetry reductions of the 3-dimensional Heisenberg distribution and 4-dimensional Engel distribution respectively. Similarly, we realise the Martinet distribution as an alternative symmetry reduction of the Engel distribution. These reductions allow us to derive…
Study geodesics on Grushin spaces, proving upper bounds on conjugate times.
problem Classify geodesics on higher-dimensional Grushin spaces.
method Solve Hamilton's equations using calculus of generalized trigonometric functions, analyze symmetries, and use density arguments.
result Prove a conjectured cut time provides an upper bound on conjugate times.
Study geometric quantum confinement on special incomplete Riemannian manifolds.
problem Characterize quantum confinement on Grushin-type manifolds.
method Constant-fibre direct integral scheme combined with Weyl's analysis.
result Fully characterizes essential self-adjointness of Laplace-Beltrami operator.
The paper characterizes Ricci solitons on the Poincaré upper half plane.
problem Characterizing Ricci solitons on the Poincaré upper half plane.
method Classifying and generalizing Ricci solitons and soliton equations in the half plane of Poincaré.
result Obtained some properties of solitons about their geodesic flows.
We show that, for all α≥0, the generalized Grushin plane Gα is bi-Lipschitz homeomorphic to a 2-dimensional quasiplane in the Euclidean space R[α]+2, where [α] is the integer part of α. The target dimension is sharp. This generalizes a recent result of Wu.
Study metrics on half plane with specific curvature properties.
problem Warped product metrics on half plane.
method Holomorphic isometries and sectional curvature analysis.
result Metrics with zero and unbounded negative curvature exist.
Classifies quantum particle behavior on a special cylinder.
problem Quantum confinement and transmission on a Grushin cylinder.
method Characterizes self-adjoint realizations of the Laplace-Beltrami operator.
result Identifies physically meaningful extensions of the Hamiltonian.
Study on migrating elastic flows of curves across half-planes.
problem Migrating elastic flows of curves from upper to lower half-planes.
method Analytical and numerical construction of migrating elastic flows.
result Construction of various migrating elastic flows.
Geodesics on extended Siegel-Jacobi upper half-plane determined.
problem Determining geodesics on a complex geometric space.
method Equating parameters in geodesic equations on the extended Siegel-Jacobi upper half-plane.
result Geodesic equations on Siegel-Jacobi, Siegel, and Heisenberg spaces.
This article deals with 2d almost Riemannian structures, which are generalized Riemannian structures on manifolds of dimension 2. Such sub-Riemannian structures can be locally defined by a pair of vector fields (X,Y), playing the role of orthonormal frame, that may become colinear on some subset. We denote D = span(X,Y…
Study shows failure of curvature-dimension conditions on sub-Riemannian manifolds.
problem Failure of curvature-dimension conditions on sub-Riemannian manifolds.
method Proves failure of curvature-dimension conditions using tangent isometries and Killing vector fields.
result Proves failure of curvature-dimension conditions on sub-Riemannian manifolds.
Study variational properties of curves in half-plane with area constraints.
problem Characterize critical points of inverse mean curvature.
method Variational analysis of curves with boundary constraints.
result Existence and stability of critical points with prescribed area.
Proves regularity for Brakke flow near stationary half-plane.
problem Regularity of Brakke flow near stationary half-planes.
method Viscosity techniques from Savin applied to Brakke's flow.
result Proves C1,α regularity for flows close to stationary half-planes. Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.
problem Understanding Berry phases and connection matrices on Siegel-Jacobi spaces.
method Examines the Siegel-Jacobi disk and upper half-plane, calculates connection matrices and covariant derivatives.
result Calculates the connection matrix and covariant derivatives on the extended Siegel-Jacobi upper half-plane.
We prove the existence of "half-plane differentials" with prescribed local data on any Riemann surface. These are meromorphic quadratic differentials with higher-order poles which have an associated singular flat metric isometric to a collection of euclidean half-planes glued by an interval-exchange map on their bounda…
Extended Siegel-Jacobi upper half-plane geometry studied with invariant metrics.
problem Characterizing the geometry of the extended Siegel-Jacobi upper half-plane.
method Parameterized using S-coordinates and expressed in terms of invariant metrics.
result Extended Siegel-Jacobi upper half-plane is a reductive, non-symmetric manifold.
The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometric…
Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 1…
We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomor…
The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.
problem Characterizing self-adjoint extensions of the Laplace-Beltrami operator on α-Grushin manifolds. method Introducing an exotic calculus of pseudodifferential operators adapted to the geometry of the singularity.
result Criterion for essential self-adjointness and determination of several self-adjoint extensions.
Study of stability conditions on 3-folds, focusing on walls and intersections.
problem Understanding stability conditions and numerical walls on 3-folds.
method Differential geometry analysis of numerical walls, proving intersections and maximum turning points.
result Gieseker semistability equivalent to asymptotic semistability along paths in the upper half plane.
In this paper, we build properly embedded singly periodic minimal surfaces which have infinite total curvature in the quotient by the period. These surfaces are constructed by adding a handle to the toroidal half-plane layers defined by H. Karcher. The technics that use is to solve a Jenkins-Serrin problem over a strip…
Study on heat flow across two half-lines with special boundary conditions.
problem Low energy mode of heat flow transmission across a Grushin-type cylinder.
method Analysis of heat equation with inverse-square potential and bridging boundary conditions.
result First insight into qualitative features of the heat flow solution at later times.
Study proves unique compactification of hyperbolic space.
problem Proving uniqueness of compactification of hyperbolic space.
method Analyzing one-parameter family of elliptic PDEs on hyperbolic space.
result Euclidean half-plane is the only compactification of hyperbolic space.
The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerat…
In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a f…
Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.
problem Isometry and eigenvalue gap in Finslerian models.
method Presented models and isometry proofs.
result First eigenvalue gapless for all three models.
The study proves sub-Riemannian manifolds cannot satisfy CD conditions unless they are Riemannian.
problem Characterizing sub-Riemannian manifolds that satisfy CD conditions. method Analysis of tangent cones and geodesics, construction of new RCD structures. result Sub-Riemannian manifolds are never CD(K,N) unless they are Riemannian. Extensions of Brownian motion to singular surfaces are studied.
problem Diffusion across singularities on surfaces.
method One-parameter family of Grushin-type singularities, heat crossing analysis, isometry group respect, Bessel processes.
result Complete description and classification of diffusions for various singularity cases.
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this p…
Study of translators in solvable group, finding new examples and non-existence.
problem Classifying translators in solvable groups invariant under isometries.
method Classification of translators in Sol3 using invariant under one-parameter group of isometries. result New graphical translators in Sol3 on half-planes, contradicting Euclidean translator rigidity. We study global Mumford-Shah minimizers in RN, introduced by Bonnet as blow-up limits of Mumford-Shah minimizers. We prove a new monotonicity formula for the energy of u when the singular set K is contained in a smooth enough cone. We then use this monotonicity to prove that for any reduced global minimizer $(u…
New Weyl's laws discovered for compact spaces with Ricci curvature bounds.
problem Understanding growth rates of eigenvalues in compact spaces with Ricci curvature constraints.
method Developed new properties of α-Grushin halfplanes and analyzed singular sets of null capacities. result Established Weyl's laws with power growth and logarithmic corrections for compact spaces.
Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of…
Study on non-classical generating sets in Fuchsian Schottky groups.
problem Estimating non-classical Schottky structure in discrete subgroups.
method Investigated Fuchsian Schottky groups with non-classical generating sets using Möbius transformations.
result Derived two non-trivial examples of Fuchsian Schottky groups with non-classical generating sets.
Constructs weight 1/2 multiplier systems for a specific group and relates to geometric edge paths.
problem Constructing weight 1/2 multiplier systems for a specific group.
method Defines an eta function and Rademacher symbol, relates to geometric edge paths in a triangulation of the upper half plane.
result Relates weight 1/2 multiplier systems to geometric edge paths.
We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields. It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric.
In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, d…