Local non-injectivity of the exponential map at critical points in sub-Riemannian geometry
(2023)
Journal Article
Borza, S., & Klingenberg, W. (2024). Local non-injectivity of the exponential map at critical points in sub-Riemannian geometry. Nonlinear Analysis: Theory, Methods and Applications, 239, Article 113421. https://doi.org/10.1016/j.na.2023.113421
Outputs (33)
Roots of Polynomials and Umbilics of Surfaces (2023)
Journal Article
Guilfoyle, D., & Klingenberg, W. (2023). Roots of Polynomials and Umbilics of Surfaces. Results in Mathematics, 78(6), https://doi.org/10.1007/s00025-023-02003-4
Regularity and Continuity properties of the sub-Riemannian exponential map (2023)
Journal Article
Borza, S., & Klingenberg, W. (2023). Regularity and Continuity properties of the sub-Riemannian exponential map. Journal of Dynamical and Control Systems, 29(4), 1385-1407. https://doi.org/10.1007/s10883-022-09624-yWe prove a version of Warner’s regularity and continuity properties for the sub-Riemannian exponential map. The regularity property is established by considering sub-Riemannian Jacobi fields while the continuity property follows from studying the Mas... Read More about Regularity and Continuity properties of the sub-Riemannian exponential map.
Weyl Estimates for spacelike hypersurfaces in de Sitter space (2022)
Journal Article
Ballesteros-Chavez, D., Klingenberg, W., & Lambert, B. (2022). Weyl Estimates for spacelike hypersurfaces in de Sitter space. Pacific journal of mathematics, 320(1), 1-11. https://doi.org/10.2140/pjm.2022.320.1We study the isometric spacelike embedding problem in scaled de Sitter space, and obtain Weyl-type estimates and the corresponding closedness in the space of embeddings.
Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows (2021)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2021). Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows. Partial Differential Equations and Applications, 2(6), Article 72. https://doi.org/10.1007/s42985-021-00128-1In the 1950’s Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper, we investigate conditions under which evolving a smooth... Read More about Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows.
Prescribed $k$ symmetric curvature hypersurfaces in de Sitter space (2020)
Journal Article
Ballesteros-Chávez, D., Klingenberg, W., & Lambert, B. (2021). Prescribed $k$ symmetric curvature hypersurfaces in de Sitter space. Canadian Mathematical Bulletin, 64(4), 886-901. https://doi.org/10.4153/s0008439520000910We prove the existence of compact spacelike hypersurfaces with prescribed k-curvature in de Sitter space, where the prescription function depends on both space and the tilt function.
Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces (2020)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2020). Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces. Annales de la Faculté des sciences de Toulouse (En ligne), 29(3), 565-576. https://doi.org/10.5802/afst.1639We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with... Read More about Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces.
A global version of a classical result of Joachimsthal (2019)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2019). A global version of a classical result of Joachimsthal. Houston journal of mathematics, 45(2), 455-467A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we... Read More about A global version of a classical result of Joachimsthal.
Mean Curvature Flow of Compact Spacelike Submanifolds in Higher Codimension (2019)
Journal Article
Guilfoyle, B., & Klingenberg, K. (2019). Mean Curvature Flow of Compact Spacelike Submanifolds in Higher Codimension. Transactions of the American Mathematical Society, 372(9), 6263-6281. https://doi.org/10.1090/tran/7766We prove the longtime existence for mean curvature flow of a smooth n-dimensional spacelike submanifold of an (n + m)-dimensional manifold whose metric satisfies the timelike curvature condition.
Parabolic Classical Curvature Flows (2017)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2018). Parabolic Classical Curvature Flows. Journal of the Australian Mathematical Society, 104(3), 338-357. https://doi.org/10.1017/s1446788717000210We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flo... Read More about Parabolic Classical Curvature Flows.
A Converging Lagrangian Flow in the Space of Oriented Line (2016)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2016). A Converging Lagrangian Flow in the Space of Oriented Line. Kyushu journal of mathematics, 70(2), 343-351. https://doi.org/10.2206/kyushujm.70.343Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the three-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of... Read More about A Converging Lagrangian Flow in the Space of Oriented Line.
Erratum to: On the geometry of the space of oriented geodesicd (2016)
Journal Article
Alekseevsky, D., Guilfoyle, B., & Klingenberg, W. (2016). Erratum to: On the geometry of the space of oriented geodesicd. Annals of Global Analysis and Geometry, https://doi.org/10.1007/s10455-016-9515-3
Totally null surfaces in neutral Kähler 4-manifolds (2016)
Journal Article
Georgiou, N., Guilfoyle, B., & Klingenberg, W. (2016). Totally null surfaces in neutral Kähler 4-manifolds. Balkan Journal of Geometry and its Applications, 21(1), 27-41We study the totally null surfaces of the neutral Kähler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (α-planes) or anti-self-dual (β-planes) and so we consider α-surfaces and β-surfaces. The metric... Read More about Totally null surfaces in neutral Kähler 4-manifolds.
On the geometry of spaces of oriented geodesics. (2011)
Journal Article
Alekseevsky, D., Guilfoyle, B., & Klingenberg, W. (2011). On the geometry of spaces of oriented geodesics. Annals of Global Analysis and Geometry, 40(4), 389-409. https://doi.org/10.1007/s10455-011-9261-5Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we de... Read More about On the geometry of spaces of oriented geodesics..
On Weingarten surfaces in Euclidean and Lorentzian 3-space (2010)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2010). On Weingarten surfaces in Euclidean and Lorentzian 3-space. Differential Geometry and its Applications, 28(4), 454-468. https://doi.org/10.1016/j.difgeo.2009.12.002We study the neutral Kähler metric on the space of time-like lines in Lorentzian View the MathML source, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric,... Read More about On Weingarten surfaces in Euclidean and Lorentzian 3-space.
On C2-smooth Surfaces of Constant Width. (2009)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2009). On C2-smooth Surfaces of Constant Width. Tbilisi Mathematical Journal, 2, 1-17In this paper, we obtain a number of results for C2-smooth surfaces of constant width in Euclidean 3-space E3-. In particular, we establish an integral inequality for constant width surfaces. This is used to prove that the ratio of volume to cubed wi... Read More about On C2-smooth Surfaces of Constant Width..
Geodesic flow on the normal congruence of a minimal surface. (2009)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2009). Geodesic flow on the normal congruence of a minimal surface
Area-stationary surfaces in neutral Kähler 4-manifolds (2008)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2008). Area-stationary surfaces in neutral Kähler 4-manifolds. Contributions to Algebra and Geometry, 49(2), 481-490
A neutral Kähler surface with applications in geometric optics. (2008)
Book Chapter
Guilfoyle, B., & Klingenberg, W. (2008). A neutral Kähler surface with applications in geometric optics. In D. V. Alekseevsky, & H. Baum (Eds.), Recent Developments in Pseudo-Riemannian Geometry (149-178). European Mathematical Society. https://doi.org/10.4171/051-1/5The space L of oriented lines, or rays, in Euclidean 3-space E3 is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kähler metric which is closely related to the Euclidean metric on E3. In thi... Read More about A neutral Kähler surface with applications in geometric optics..
Geodesic Flow on Global Holomorphic Sections of TS^2 (2007)
Journal Article
Klingenberg, W., & Guilfoyle, B. (2007). Geodesic Flow on Global Holomorphic Sections of TS^2. Bulletin of the Belgian Mathematical Society Simon Stevin (Printed), 14(2), 363-371We study the geodesic flow on the global holomorphic sections of the bundle π:TS2→S2π:TS2→S2 induced by the neutral Kähler metric on the space of oriented lines of R3R3, which we identify with TS2TS2. This flow is shown to be completely integrable wh... Read More about Geodesic Flow on Global Holomorphic Sections of TS^2.
On Hamilton's characteristic functions for reflection (2006)
Journal Article
Klingenberg, W., & Guilfoyle, B. (2006). On Hamilton's characteristic functions for reflection. Bulletin - Irish Mathematical Society, 57, 29-40
Reflection in a translation invariant surface (2006)
Journal Article
surface. Mathematical Physics, Analysis and Geometry, 9(3), 225-231. https://doi.org/10.1007/s11040-007-9013-8We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, w... Read More about Reflection in a translation invariant surface.
Isolated umbilic points on surfaces in R^3 (2006)
Journal Article
Klingenberg, W., & Guilfoyle, B. (2006). Isolated umbilic points on surfaces in R^3. Deltio tīs Ellīnikīs mathīmatikīs etaireias (1960), 51, 23-30
Reflection of a wave off a surface. (2006)
Journal Article
Klingenberg, W., & Guilfoyle, B. (2006). Reflection of a wave off a surface. Journal of Geometry, 84(1), 55-72. https://doi.org/10.1007/s00022-005-0022-0Recent investigations of the space of oriented lines in R3R3 are applied to geometric optics. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecti... Read More about Reflection of a wave off a surface..
Level sets of functions and symmetry sets of surface sections (2005)
Presentation / Conference Contribution
Diatta, A., Giblin, P., Guilfoyle, B., & Klingenberg, W. (2005). Level sets of functions and symmetry sets of surface sections. In R. . R. Martin, H. . E. Bez, & M. . A. Sabin (Eds.), Mathematics of surfaces XI: 11th IMA international conference, Loughborough, UK, September 5-7, 2005: proceedings (147-160). https://doi.org/10.1007/11537908_9We prove that the level sets of a real C s function of two variables near a non-degenerate critical point are of class C [s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an ellipti... Read More about Level sets of functions and symmetry sets of surface sections.
The Casimir effect between non-parallel plates by geometric optics (2005)
Journal Article
Guilfoyle, B., Klingenberg, W., & Sen, S. (2005). The Casimir effect between non-parallel plates by geometric optics. Reviews in Mathematical Physics, 17(8), 859 - 880. https://doi.org/10.1142/s0129055x05002431The first two authors have developed a technique which uses the complex geometry of the space of oriented affine lines in ℝ3 to describe the reflection of rays off a surface. This can be viewed as a parametric approach to geometric optics which has m... Read More about The Casimir effect between non-parallel plates by geometric optics.
An indefinite Kaehler metric on the space of oriented lines (2005)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2005). An indefinite Kaehler metric on the space of oriented lines. Journal of the London Mathematical Society, 72(2), 497-509. https://doi.org/10.1112/s0024610705006605The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space ${\mathbb{T}}$ of oriented affine lines in ${\mathbb{R}}^3$ with the tangent bundle of $S^2$. Thus the round met... Read More about An indefinite Kaehler metric on the space of oriented lines.
Generalized surfaces in R^3 (2004)
Journal Article
Klingenberg, W., & Guilfoyle, B. (2004). Generalized surfaces in R^3. Mathematical Proceedings of the Royal Irish Academy, 104A(2), 199-209
On the space of oriented affine lines in R^3 (2004)
Journal Article
Guilfoyle, B., & Klingenberg, W. (2004). On the space of oriented affine lines in R^3. Archiv der Mathematik, 82(1), 81 - 84. https://doi.org/10.1007/s00013-003-4861-3We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean and the tangent bundle to the 2-sphere. These can be utilised to give canonical coordinates on surfaces in, as we illustrate wi... Read More about On the space of oriented affine lines in R^3.
Real hypersurfaces of Kahler manifolds (2001)
Journal Article
Klingenberg, W. (2001). Real hypersurfaces of Kahler manifolds. Asian Journal of Mathematics, 5(1), 1 -18. https://doi.org/10.4310/ajm.2001.v5.n1.a1Building on work by S. M. Webster \ref[J. Differential Geom. 13 (1978), no. 1, 25--41; MR0520599 (80e:32015)] the author studies the geometry of the second fundamental form of a real hypersurface in a Kähler manifold. As an application he proves that... Read More about Real hypersurfaces of Kahler manifolds.
Flow of a hypersurface by the trace of the Levi form (1999)
Journal Article
Klingenberg, W., & Huisken, G. (1999). Flow of a hypersurface by the trace of the Levi form. Mathematical Research Letters, 6(5), 645-661
Moduli space of holomorphic discs with boundary. (1997)
Journal Article
Klingenberg, W. (1997). Moduli space of holomorphic discs with boundary. Archiv der Mathematik, 69(3), 249-253. https://doi.org/10.1007/s000130050117We study the moduli problem for holomorphic discs with boundary constrained to be in a maximally real submanifold and establish smooth dependence of the associated moduli space on the boundary condition.
Asymptotic curves on real analytic surfaces in C^2 (1985)
Journal Article
Klingenberg, W. (1985). Asymptotic curves on real analytic surfaces in C^2. Mathematische Annalen, 273(1), 149-162. https://doi.org/10.1007/bf01455920