The Sard conjecture is proven for smooth Martinet surfaces and under certain conditions for singular ones.
problem Investigating the set of points reachable by singular horizontal paths on Martinet surfaces.
method Control of divergence of vector fields and techniques of resolution of singularities.
result The Sard conjecture holds true for smooth Martinet surfaces and under specific conditions for singular ones.
Two flat sub-Lorentzian problems on Martinet distribution differ in attainable set intersections.
problem Flat sub-Lorentzian structures on Martinet distribution.
method Analysis of attainable sets, optimal trajectories, sub-Lorentzian distances and spheres.
result The attainable set for the first problem intersects with the Martinet plane, while for the second it does not.
Paper studies viscosity solutions in unique Martinet spaces.
problem Properties of viscosity solutions in Martinet spaces.
method Established properties and proved uniqueness of solutions.
result Uniqueness of viscosity solutions in Martinet spaces.
Study local invariants of singular symplectic forms on manifolds.
problem Identify local invariants of singular symplectic forms on manifolds.
method Analytic and smooth categories; structural stability; Martinet hypersurface; kernel of ω^(n-1).
result Conditions to determine the kernel of ω^(n-1) at a point by other invariants.
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…
This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a deta…
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…
We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R^2 x R^2. We introduce the notion of a finite type singularity analogous to CR geometry and, al…
We derive simple forms for saddle-node singular points of analytic foliations in the real or complex plane just by gluing foliated complex manifolds. We give the versal analytic deformation of the simplest model. We also derive a unique analytic form for those saddle-node having a central manifold. By this way, we reco…
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of point…
The abstract discusses how sub-Riemannian manifolds can have branching geodesics.
problem The existence of branching geodesics in sub-Riemannian geometry.
method Analyzing the rank discontinuity of normal geodesics and constructing specific examples.
result Sub-Riemannian manifolds can contain branching normal minimizing geodesics.
Contractibility of tight contact structures on S3 proven.
problem Proving the contractibility of the space of tight contact structures on S3. method Using methods from the first author's paper on contact 3-manifolds.
result The space of tight contact structures on S3 is contractible. Study on contact Hamiltonian functions for singular contact structures.
problem Understanding infinitesimal contact transformations on singular contact structures.
method Showed injectivity and provided an explicit local formula for the inverse map.
result Explicit local formula for the inverse map when contact structure has singularities of the first type.
New trigonometric method for convex sets aids control problems.
problem Optimal control problems with two-dimensional control.
method Proposes a new method to describe convex sets.
result Investigates sub-Finsler problems in various groups.
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma z calculus and z--Bochner's formula. result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
Study on convergence rates of degenerate SDEs using Fisher information and generalized Bochner's formula.
problem Analysis of dynamical behaviors of degenerate stochastic differential equations.
method Use of Fisher information as Lyapunov functional, generalized Gamma calculus, and generalized Bochner's formula.
result Derivation of convergence rate conditions and examples in specific sub-Riemannian structures.
We present a new optimal systolic inequality for a closed Riemannian manifold X, which generalizes a number of earlier inequalities, including that of C. Loewner. We characterize the boundary case of equality in terms of the geometry of the Abel-Jacobi map, A_X, of X. For an extremal metric, the map A_X turns out to be…
In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact plus or minus 1 surgeries, and used this to prove that any (closed) contact…
Study spectral properties of sub-Riemannian Laplacians, proving quantum ergodicity and heat kernel asymptotics.
problem Spectral properties of sub-Riemannian Laplacians.
method Quantum ergodicity results, small-time asymptotics of sub-Riemannian heat kernels, Weyl law.
result Weyl law and spectral concentration on Lie brackets of length r-1.
New sub-Riemannian structures fail synthetic curvature bounds.
problem Failure of synthetic curvature bounds in sub-Riemannian geometry.
method New stability results for local MCP under quotients, applied to specific sub-Riemannian structures.
result Ideal sub-Riemannian structures can fail the MCP, generically for high dimensions and rank > 3.
The paper analyzes covariate shift in nonparametric regression with Markovian data.
problem Covariate shift in regression problems with Markovian data.
method Extension of nonparametric convergence rates to Markovian dependence structures, using Hölder smoothness assumptions and similarity measures.
result Precise convergence rates for Nadaraya-Watson kernel estimators under specific Markovian conditions.
In this paper we prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure ass…
New findings show transfer learning is possible even when density ratios are unbounded.
problem Transfer learning under unbounded density ratios.
method Low-degree polynomial estimators, general transfer inequality over Rn. result Non-trivial transfer learning possible under mild assumptions, including log-concave measures.
Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.
problem Limiting the outcomes of gluing Scherk surfaces into minimal surfaces.
method Constructing minimal surfaces by stacking and gluing doubly periodic Scherk surfaces.
result Except for special cases, gluing more Scherk surfaces results in known minimal surfaces.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
The study characterizes polynomial conserved quantities for Lie applicable surfaces.
problem Characterizing polynomial conserved quantities for Lie applicable surfaces.
method Gauge theoretic approach for Lie applicable surfaces, including isothermic, Guichard, and L-isothermic surfaces. result Induced transformations of Lie applicable surfaces for well-known transformations and new Bäcklund-type transformation for linear Weingarten surfaces.
Study on focal surfaces of tubular surfaces in 3D space, focusing on their flatness and asymptotic properties.
problem Characterizing and understanding focal surfaces of tubular surfaces in 3D space.
method Defined tubular surfaces using Frenet and Darboux frames, analyzed their focal surfaces, and derived conditions for flatness.
result No minimal focal surface exists in 3D space for tubular surfaces.
Generalizes ribbonness result for surface-links.
problem Characterizing ribbon surface-links.
method Analyzes handle-irreducible summands and uses equivalences.
result Every stable-ribbon surface-link is a ribbon surface-link.
New higher-dimensional Schwarz and Scherk surfaces discovered.
problem Constructing complete embedded periodic minimal hypersurfaces.
method Higher dimensional generalizations of Schwarz's and Scherk's surfaces.
result Complete embedded periodic minimal hypersurfaces constructed in Rn. Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
The paper studies special surfaces with a new type of support function.
problem Characterizing surfaces with a specific quadratic support function.
method Developed a Weierstrass type representation involving holomorphic functions.
result Classified surfaces of rotation with this new type of support function.
Minimal Legendrian surfaces found in 5D sphere.
problem Characterizing Willmore Legendrian surfaces in S5. method Analyzing properties of Willmore and csL Willmore surfaces.
result Complete Willmore Legendrian surfaces in S5 are minimal. The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.
problem Understanding the isotopy and closure properties of knitted surfaces and surface-links.
method Analyzing the structure and closure of knitted surfaces and surface-links in R4. result Any surface-link is ambient isotopic to the closure of a 2-dimensional knit.
Classifies surfaces with constant Gaussian curvature in Euclidean 3-space.
problem Classifying surfaces with constant Gaussian curvature in Euclidean 3-space.
method Analyzing surfaces as implicit equations and proving properties based on Gaussian curvature.
result Surfaces with constant Gaussian curvature are either surfaces of revolution, cylindrical surfaces, conical surfaces, or have specific forms.
We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces with constant mean curvature one (CMC-1 surfaces), HMC-1 surfaces belong to a cert…
Unstable minimal surfaces in n-space link to hyperbolic products.
problem Characterizing unstable minimal surfaces in Rn and their product counterparts. method Lifting to R-trees, deforming to hyperbolic products, and proving instability equivalence. result Unstable minimal surfaces in Rn imply unstable surfaces in product hyperbolic spaces. Researchers generalize Ribaucour-type surfaces with new mathematical representation.
problem Defining and characterizing new geometric surfaces.
method Developed a new mathematical representation for GRT-surfaces involving holomorphic functions and a real function.
result Explicit examples and classification of GRT-surfaces of rotation.
Study on dual surfaces of rotational minimal and maximal in Euclidean and Lorentz-Minkowski spaces.
problem Investigating the duality between minimal and maximal surfaces in different spaces.
method Analysis of rotational surfaces and use of one-parameter group of rotations.
result Family of Bonnet minimal and maximal surfaces emerge in the duality process.
Stable-ribbon surface-links have unique handle-irreducible summands.
problem Characterize handle-irreducible summands of stable-ribbon surface-links.
method Combining old results on stably trivial surface-links and surface-knots with infinite cyclic fundamental groups.
result Every handle-irreducible summand of a stably trivial surface-link is a trivial 2-link.
Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.
problem Understanding automorphisms of fine curve graphs on surfaces.
method Analyzing vertices and edges of fine curve graphs to match with surface homeomorphisms.
result Automorphism group of fine curve graphs is naturally isomorphic to the homeomorphism group of boundaryless planar surfaces with at least 7 punctures.
This paper connects Laguerre minimal surfaces to Weierstrass representations.
problem Understanding the relationship between Laguerre minimal surfaces and Weierstrass representations.
method Defining spherical mean curvature and providing Weierstrass-type representations for two classes of surfaces.
result Laguerre minimal surfaces are related to H2-surfaces, providing a new Weierstrass-type representation. Paper uses harmonic surfaces to model surfaces of any complexity.
problem Topological limitations of minimal surfaces restrict their use.
method Weierstrass representation for harmonic surfaces to construct arbitrary genus surfaces.
result Harmonic surfaces can be used to create surfaces of arbitrary genus.
New surfaces in 4-ball constructed from knits, described by charts.
problem Constructing surfaces in 4-ball from knits.
method Introducing knitted surfaces, describing them with BMW charts.
result Every compact surface in 4-ball is ambiently isotopic to a knitted surface.
Study on singular points of translation surfaces under linearly dependent conditions.
problem Investigate singular points of translation surfaces under linearly dependent conditions.
method Use theories of generalised framed surfaces and framed surfaces.
result Introduce translation generalised framed surfaces and investigate their singular points.
Crochet patterns for minimal surfaces created using trigonometry.
problem Creating crochet patterns for minimal surfaces.
method Using trigonometric identities to calculate arc lengths.
result Crochet instructions for Enneper's surface.
Super Riemann surfaces extend Riemann surfaces with an additional field, the gravitino.
problem Extending the study of Riemann surfaces to include supergeometry.
method Presenting an extension of the harmonic action functional to super Riemann surfaces.
result Super Riemann surfaces can be studied using an extended harmonic action functional.
New surfaces generalize Dini surfaces in 4D.
problem None explicitly stated; focuses on surface generalization.
method Introducing a new family of surfaces in 4D.
result Generalized Dini surfaces exist in 4D.