Ansattprofil

Erlend Grong

Professor II

Campus Bergen, Institutt for samfunnsøkonomi

E-post erlend.grong@bi.no

Publikasjoner

Vitenskapelige publikasjoner

Andre

Viser 5 av 37 publikasjon(er)

Artikkel Erlend Grong (2026)

Canonical connections on sub-Riemannian manifolds with constant symbol

Analysis and Mathematical Physics 16(2) Doi: https://doi.org/10.1007/s13324-025-01155-6

As a tool to address the equivalence problem in sub-Riemannian geometry, we introduce a canonical choice of grading and compatible affine connection, available on any sub-Riemannian manifold with constant symbol. This grading and affine connection is based on Morimoto’s normalization conditions in [30]. We completely compute these structures for contact manifolds of constant symbol, including the cases where the connections of Tanaka-Webster-Tanno are not defined. We also give an original intrinsic grading on sub-Riemannian (2,3,5)-manifolds, and use this to present the first flatness theorem in this setting.

Artikkel Erlend Grong, Sylvie Vega-Molino (2026)

Controllability on landmark manifolds for shapes and neural ODEs

MCSS. Mathematics of Control, Signals and Systems Doi: https://doi.org/10.1007/s00498-026-00435-1

Landmark manifolds consist of finite collections of distinct points in an underlying space, referred to as landmark configurations. The dynamics of these configurations can be used to represent flows, such as solutions to ODEs or shape deformations. In this work, we consider landmark configurations in Euclidean space and study how they can be connected via flows of vector fields. For dimensions greater than or equal to two, we explicitly construct two vector fields whose flows can connect any pair of landmark configurations with the same cardinality. This property is known as exact universal interpolation. In dimension one, we show that the same result holds for pairs of landmark configurations that share the same relative ordering. In all dimensions, controllability is achieved using one constant vector field and one polynomial vector field of degree three.

Artikkel Erlend Grong, Jorge Hidalgo, Sylvie Vega-Molino (2025)

A sub-Riemannian Gauss-Bonnet theorem for surfaces in contact manifolds

Journal of Geometric Analysis 35(8) Doi: https://doi.org/10.1007/s12220-025-02076-3

We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of three-dimensional contact sub-Riemannian manifolds. Using a family of taming Riemannian metrics on the surface which approach a sub-Riemannian metric, we carefully study the usual Gauss-Bonnet formula under this limit. We are then able to recover the Euler characteristic of the surface from the geometry around the surface’s characteristic set, i.e., the points where the tangent space to the surface and contact structure coincide. For the case of surfaces with boundary, we also give a Gauss-Bonnet type result which includes limits related to the boundary’s curvature.

Artikkel Francesco Ballerin, Erlend Grong (2025)

Geometry of the visual cortex with applications to image inpainting and enhancement

Graphical Models 137 Doi: https://doi.org/10.1016/j.gmod.2024.101239

Equipping the rototranslation group SE(2) with a sub-Riemannian structure inspired by the visual cortex V1, we propose algorithms for image inpainting and enhancement based on hypoelliptic diffusion. We innovate on previous implementations of the methods by Citti, Sarti, and Boscain et al., by proposing an alternative that prevents fading and is capable of producing sharper results in a procedure that we call WaxOn-WaxOff. We also exploit the sub-Riemannian structure to define a completely new unsharp filter using SE(2), analogous to the classical unsharp filter for 2D image processing. We demonstrate our method on blood vessels enhancement in retinal scans.

Artikkel Fabrice Baudoin, Erlend Grong, Luca Rizzi, Sylvie Vega-Molino (2025)

Comparison theorems on H-type sub-Riemannian manifolds

Calculus of Variations and Partial Differential Equations 64(5) Doi: https://doi.org/10.1007/s00526-025-02992-w

On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet–Myers theorem that extends to this general setting results previously proved on contact and quaternionic contact manifolds.

Artikkel Erlend Grong, Alexander Schmeding (2024)

Controllability and diffeomorphism groups on manifolds with boundary

Forum mathematicum Doi: https://doi.org/10.1515/forum-2024-0160

In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map.

Artikkel Erlend Grong, Hans Zanna Munthe-Kaas, Jonatan Stava (2024)

Post-Lie Algebra Structure of Manifolds with Constant Curvature and Torsion

Journal of Lie theory 34(2) s. 339-352

Artikkel Erlend Grong, Stefan Sommer (2024)

Most Probable Flows for Kunita SDEs

Applied Mathematics and Optimization 89(2) Doi: https://doi.org/10.1007/s00245-024-10110-z

We identify most probable flows for Kunita Brownian motions, i.e. stochastic flows with Eulerian noise and deterministic drifts. Such stochastic processes appear for example in fluid dynamics and shape analysis modelling coarse scale deterministic dynamics together with fine-grained noise. We treat this infinite dimensional problem by equipping the underlying domain with a Riemannian metric originating from the noise. The resulting most probable flows are compared with the non-perturbed deterministic flow, both analytically and experimentally by integrating the equations with various choice of noise structures.

Artikkel Erlend Grong, Irina Markina (2023)

Harmonic maps into sub-Riemannian Lie groups

Communications in analysis and mechanics 15(3) s. 515-532 Doi: https://doi.org/10.3934/cam.2023025

Artikkel Fabrice Baudoin, Erlend Grong, Luca Rizzi, Gianmarco Vega-Molino (2022)

H-type foliations

Differential geometry and its applications 85:101952 Doi: https://doi.org/10.1016/j.difgeo.2022.101952

With a view toward sub-Riemannian geometry, we introduce and study H-type foliations. These structures are natural generalizations of K-contact geometries which encompass as special cases K-contact manifolds, twistor spaces, 3K-contact manifolds and H-type groups. Under an horizontal Ricci curvature lower bound on these structures, we prove a sub-Riemannian diameter upper bounds and first eigenvalue estimates for the sub-Laplacian. Then, using a result by Moroianu-Semmelmann [38], we classify the H-type foliations that carry a parallel horizontal Clifford structure. Finally, we prove an horizontal Einstein property and compute the horizontal Ricci curvature of these spaces in codimension more than 2.

Artikkel Erlend Grong (2022)

Curvature and the equivalence problem in sub-Riemannian geometry

Archivum Mathematicum 58(5) s. 295-327 Doi: https://doi.org/10.5817/AM2022-5-295

Artikkel Erlend Grong, Stefan Sommer (2022)

Most Probable Paths for Anisotropic Brownian Motions on Manifolds

Foundations of Computational Mathematics Doi: https://doi.org/10.1007/s10208-022-09594-4

Artikkel Fabrice Baudoin, Erlend Grong, Gianmarco Vega-Molino (2022)

A horizontal Chern–Gauss–Bonnet formula on totally geodesic foliations

Annals of Global Analysis and Geometry 61 s. 759-776 Doi: https://doi.org/10.1007/s10455-022-09827-3

Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors in Baudoin and Grong (Ann Glob Anal Geom 56(2):403–428, 2019).

Artikkel Erlend Grong, Torstein Nilssen, Alexander Schmeding (2022)

Geometric rough paths on infinite dimensional spaces

Journal of Differential Equations 340 s. 151-178 Doi: https://doi.org/10.1016/j.jde.2022.08.034

Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For α ∈ (1/3,1/2), we give criteria for when we can approximate Banach space-valued weakly geometric α-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.

Artikkel Eirik Berge, Erlend Grong (2021)

On G2 and sub-Riemannian model spaces of step and rank three

Mathematische Zeitschrift 65 s. 1853-1885 Doi: https://doi.org/10.1007/s00209-020-02653-y

We give the complete classification of all sub-Riemannian model spaces with both step and rank three. Model spaces in this context refer to spaces where any infinitesimal isometry between horizontal tangent spaces can be integrated to a full isometry. They will be divided into three families based on their nilpotentization. Each family will depend on a different number of parameters, making the result crucially different from the known case of step two model spaces. In particular, there are no nontrivial sub-Riemannian model spaces of step and rank three with free nilpotentization. We also realize both the compact real form gc2 and the split real form gs2 of the exceptional Lie algebra g2 as isometry algebras of different model spaces.

Artikkel Erlend Grong (2021)

Model spaces in sub-Riemannian geometry

Communications in analysis and geometry 29(1) s. 77-113 Doi: https://doi.org/10.4310/CAG.2021.V29.N1.A3

Artikkel Li-Juan Cheng, Erlend Grong, Anton Thalmaier (2021)

Functional inequalities on path space of sub-Riemannian manifolds and applications

Nonlinear Analysis 210 Doi: https://doi.org/10.1016/j.na.2021.112387

We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians of a sub-Riemannian structure with a chosen complement. We introduce a concept of gradient for cylindrical functionals on path space in such a way that the gradient operators are closable in . With this structure in place, we show that a bound on horizontal Ricci curvature is equivalent to several inequalities for functions on path space, such as a gradient inequality, log-Sobolev inequality and Poincaré inequality. As a consequence, we also obtain a bound for the spectral gap of the Ornstein–Uhlenbeck operator.

Artikkel Fabrice Baudoin, Erlend Grong, Kazumasa Kuwada, Robert Neel, Anton Thalmaier (2020)

Radial processes for sub-Riemannian Brownian motions and applications

Electronic Journal of Probability (EJP) 25 Doi: https://doi.org/10.1214/20-EJP501

We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô’s formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng’s type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.

Artikkel Yacine Chitour, Erlend Grong, Frederic Jean, Petri Kokkonen (2019)

Horizontal holonomy and foliated manifolds

Annales de l'Institut Fourier 69(3) s. 1047-1086 Doi: https://doi.org/10.5802/aif.3265

We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle D of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose–Singer’s and Ozeki’s theorems. We then give necessary and sufficient conditions in terms of the horizontal holonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure. The subbundle D plays the role of an orthogonal complement to the leaves of the foliation in case (a) and of a principal connection in case (b).

Artikkel Mauricio Godoy Molina, Erlend Grong, Irina Markina (2019)

Submersions and curves of constant geodesic curvature

Mathematische Nachrichten 292(9) s. 1956-1971 Doi: https://doi.org/10.1002/mana.201800352

Artikkel Stine Marie Berge, Erlend Grong (2019)

A Lichnerowicz estimate for the spectral gap of a sub-Laplacian

Proceedings of the American Mathematical Society 147(12) s. 5153-5166 Doi: https://doi.org/10.1090/proc/14223

Abstract: For a second order operator on a compact manifold satisfying the strong Hörmander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.

Artikkel Fabrice Baudoin, Erlend Grong (2019)

Transverse Weitzenbock formulas and de Rham cohomology of totally geodesic foliations

Annals of Global Analysis and Geometry 56(2) s. 403-428 Doi: https://doi.org/10.1007/s10455-019-09672-x

Artikkel Fabrice Baudoin, Erlend Grong, Kazumasa Kuwada, Anton Thalmaier (2019)

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

Calculus of Variations and Partial Differential Equations 58(4) Doi: https://doi.org/10.1007/s00526-019-1570-8

Artikkel Erlend Grong, Pierre Pansu (2018)

Asymptotic expansion of holonomy.

Forum mathematicum 30(6) s. 1363-1385 Doi: https://doi.org/10.1515/forum-2017-0029

Artikkel Erlend Grong, Anton Thalmaier (2018)

Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds

Potential Analysis 51(2) s. 219-254 Doi: https://doi.org/10.1007/s11118-018-9710-x

Artikkel Mauricio Godoy Molina, Erlend Grong (2017)

Riemannian and sub-Riemannian geodesic flows.

Journal of Geometric Analysis 27(2) s. 1260-1273 Doi: https://doi.org/10.1007/s12220-016-9717-8

Artikkel Erlend Grong (2016)

Submersions, Hamiltonian systems, and optimal solutions to the rolling manifolds problem.

SIAM Journal of Control and Optimization 54(2) s. 536-566 Doi: https://doi.org/10.1137/15m1008919

Artikkel Erlend Grong, Anton Thalmaier (2016)

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part I.

Mathematische Zeitschrift 282(1-2) s. 99-130 Doi: https://doi.org/10.1007/s00209-015-1534-4

Artikkel Erlend Grong, Anton Thalmaier (2016)

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part II.

Mathematische Zeitschrift 282(1-2) s. 131-164 Doi: https://doi.org/10.1007/s00209-015-1535-3

Artikkel Erlend Grong, Irina Markina, Alexander Vasiliev (2015)

Sub-Riemannian Geometry on Infinite-Dimensional Manifolds

Journal of Geometric Analysis 25(4) s. 2474-2515 Doi: https://doi.org/10.1007/s12220-014-9523-0

Artikkel Mauricio Godoy Molina, Erlend Grong (2014)

Geometric conditions for the existence of a rolling without twisting or slipping.

Communications on Pure and Applied Analysis 13(1) s. 435-452

Artikkel Erlend Grong (2012)

Controllability of rolling without twisting or slipping in higher dimensions

SIAM Journal of Control and Optimization 50(4) s. 2462-2485 Doi: https://doi.org/10.1137/110829581

Artikkel Mauricio Godoy Molina, Erlend Grong, Irina Markina, Fátima Silva Leite (2012)

An intrinsic formulation of the problem on rolling manifolds

Journal of Dynamical and Control Systems 18(2) s. 181-214 Doi: https://doi.org/10.1007/s10883-012-9139-2

Kapittel Erlend Grong, Irina Markina, Alexander Vasiliev (2012)

Sub-Riermannian structures corresponding to Kählerian metrics on the universal Teichmüller space and curve

60 years of Analytic Functions in Lublin. In memory of our professors and friends Jan G. Krzyz, Zdzislaw Lewandowski and Wojciech Szapiel s. 97-116

Artikkel Erlend Grong, Alexander Vasiliev (2011)

Sub-Riemannian and sub-Lorentzian geometry on SU(1,1) and on its universal cover

Journal of Geometric Mechanics (JGM) 3(2) s. 225-260 Doi: https://doi.org/10.3934/jgm.2011.3.225

Kapittel Mauricio Godoy Molina, Erlend Grong, Fátima Silva Leite, Irina Markina (2011)

Rolling manifolds : an intrinsic perspective

Textos de matemática : mathematical papers in honour of Fátima Silva Leite s. 71-84

Artikkel Erlend Grong, Pavel Gumenyuk, Alexander Vasiliev (2009)

Matching univalent functions and conformal welding

Annales Academiae Scientiarum Fennicae Mathematica 34(1) s. 303-314

Given a conformal mapping $f$ of the unit disk $\mathbb D$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of the exterior of the unit disk $\mathbb D^*$ onto the exterior domain $D^*$ regarding to $D$. The answer is expressed in terms of a linear differential equation with a driving term given as the kernel of an operator dependent on the original mapping $f$. Examples are provided. This study is related to the problem of conformal welding and to representation of the Virasoro algebra in the space of univalent functions. 2000 Mathematics Subject Classification: Primary 30C35; Secondary 17B68. Key words: Matching univalent functions, conformal welding, Kirillov's manifold, Virasoro algebra.

Viser 5 av 27 publikasjon(er)

Konferanseforedrag Erlend Grong (2021)

Akademisk grad
År	Akademisk institusjon	Grad
--N/A--	NA	Other

Erlend Grong

Publikasjoner

Canonical connections on sub-Riemannian manifolds with constant symbol

Controllability on landmark manifolds for shapes and neural ODEs

A sub-Riemannian Gauss-Bonnet theorem for surfaces in contact manifolds

Geometry of the visual cortex with applications to image inpainting and enhancement

Comparison theorems on H-type sub-Riemannian manifolds

Controllability and diffeomorphism groups on manifolds with boundary

Post-Lie Algebra Structure of Manifolds with Constant Curvature and Torsion

Most Probable Flows for Kunita SDEs

Harmonic maps into sub-Riemannian Lie groups

H-type foliations

Curvature and the equivalence problem in sub-Riemannian geometry

Most Probable Paths for Anisotropic Brownian Motions on Manifolds

A horizontal Chern–Gauss–Bonnet formula on totally geodesic foliations

Geometric rough paths on infinite dimensional spaces

On G2 and sub-Riemannian model spaces of step and rank three

Model spaces in sub-Riemannian geometry

Functional inequalities on path space of sub-Riemannian manifolds and applications

Radial processes for sub-Riemannian Brownian motions and applications

Horizontal holonomy and foliated manifolds

Submersions and curves of constant geodesic curvature

A Lichnerowicz estimate for the spectral gap of a sub-Laplacian

Transverse Weitzenbock formulas and de Rham cohomology of totally geodesic foliations

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

Asymptotic expansion of holonomy.

Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds

Riemannian and sub-Riemannian geodesic flows.

Submersions, Hamiltonian systems, and optimal solutions to the rolling manifolds problem.

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part I.

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part II.

Sub-Riemannian Geometry on Infinite-Dimensional Manifolds

Geometric conditions for the existence of a rolling without twisting or slipping.

Controllability of rolling without twisting or slipping in higher dimensions

An intrinsic formulation of the problem on rolling manifolds

Sub-Riermannian structures corresponding to Kählerian metrics on the universal Teichmüller space and curve

Sub-Riemannian and sub-Lorentzian geometry on SU(1,1) and on its universal cover

Rolling manifolds : an intrinsic perspective

Matching univalent functions and conformal welding

Path space on sub-Riemannian manifolds.

Geometric rough paths on Hilbert spaces

Geometric rough paths on Hilbert spaces.

Geometric rough paths on Hilbert spaces.

Asymptotic expansions of holonomy

Hamiltonians and Geometry

Curvature of sub-Riemannian manifolds.

Sub-Laplacians on forms and cohomology

Comparison theorems for the sub-Laplacians.

Sub-Laplacians and the curvature dimension inequality

Asymptotic expansions of holonomy

Rumin complexes and the Laplacian on forms

Horizontal holonomy and foliations

Horizontal holonomy with applications

Sub-Riemannian manifolds with a metric-preserving complement and their diffusions

Optimal solutions to the rolling problem

Infinite dimensional sub-Riemannian geometry

PDEs as geodesic equations in infinite dimensional sub-Riemannian geometry

Geometry and the problem of rolling manifolds

Frame Bundles, Rolling and Geometry.

Sub-Riemannian geometry of the group of diffeomorphisms of the circle.

Sub-Riemannian geometry of the group of diffeomorphisms of the circle

Intrinsic Rolling of Manifolds

Sub-Riemannian and sub-Lorientzian geometry on SU(1,1)

Sub-Riemannian and sub-Lorientzian geometry on SU(1,1)

Sub-Riemannian and sub-Lorientzian geometry on SU(1,1)

Optimal control