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Wilhelm Klingenberg

Publications and source records attributed to Wilhelm Klingenberg.

At least 19 recordsLinked to original sources
arXiv2026

The Bishop family of holomorphic discs: regularity and higher index

We prove $C^{k/2,\alpha/2}$ -regularity up to a non-umbilic elliptic complex point for the Bishop family of holomorphic discs with boundary in a $C^{k,\alpha}$ regular real surface. Furthermore, we prove existence and regularity of holomorphic discs near certain complex points of index $\ge 2$. The proof employs a novel blow-up of the real surface which resolves the complex point to a pair of totally real surfaces and leads to a $\mathbb{Z}_2$ -equivariant Riemann-Hilbert problem for holomorphic annuli. The index is computed to be 1 and the problem is shown to be Fredholm-regular.

math.CV↗
arXiv2026

Smirnov Decomposition of a Horizontal Vector Charge in the Heisenberg Group

A divergence-free horizontal vector current in Heisenberg space may be viewed as an element of the dual space of horizontal test vector fields. By applying a horizontal Liouville theorem in this setting, the flow lines of such a vector field generate a family of horizontal curves and an associated measure on this collection. In this paper, we provide a direct proof of the Smirnov decomposition for a Federer-Fleming current within the horizontal distribution.

math.FA↗
arXiv2025

Optimal transport on the sub-Lorentzian Heisenberg group

We investigate the synthetic metric spacetime structure of the sub-Lorentzian Heisenberg group and we study the optimal transport problem in this space. The sub-Lorentzian version of Brenier's theorem is established in this setting. Finally, we provide examples of optimal transport maps and derive a sub-Lorentzian Monge-Amp\`ere equation.

math.MG↗
arXiv2025

The three obdurate conjectures of differential geometry

We explore the role of symmetry in three obdurate conjectures of differential geometry: the Carath\'eodory, the Willmore and the Lawson Conjectures. All three Conjectures concern surfaces in 3-dimensional space-forms, which have a high degree of symmetry. It is shown that this symmetry is broken and more general ambient metrics are considered, none of the Conjectures continue to hold. The subtle manner in which symmetry enters the first Conjecture is also explained in detail.

math.DG↗
arXiv2023

Parabolic evolution with boundary to the Bishop family of holomorphic discs

It is proven that a definite graphical rotationally symmetric line congruence evolving under mean curvature flow with respect to the neutral Kaehler metric in the space of oriented lines of Euclidean 3-space, subject to suitable Dirichlet and Neumann boundary conditions, converges to a maximal surface. When the Neumann condition implemented is that the flowing disc be holomorphic at the boundary, it is proven that the flow converges to a holomorphic disc. This is extended to the flow of a family of discs with boundary lying on a fixed rotationally symmetric line congruence, which is shown to converge to a filling by maximal surfaces. Moreover, if the family is required to be holomorphic at the boundary, it is shown that the flow converges to the Bishop filling by holomorphic discs of an isolated complex point of Maslov index 2.

math.DG↗
arXiv2023

A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

Consider a convex cone in three-dimensional Minkowski space which either contains the lightcone or is contained in it. This work considers mean curvature flow of a proper spacelike strictly mean convex disc in the cone which is graphical with respect to its rays. Its boundary is required to have constant intersection angle with the boundary of the cone. We prove that the corresponding parabolic boundary value problem for the graph admits a solution for all time which rescales to a self-similarly expanding solution.

math.DG↗
arXiv2022

Local non-injectivity of the exponential map at critical points in sub-Riemannian geometry

We prove that the sub-Riemannian exponential map is not injective in any neighbourhood of certain critical points. Namely that it does not behave like the injective map of reals given by $f(x) = x^3$ near its critical point $x = 0$. As a consequence, we characterise conjugate points in ideal sub-Riemannian manifolds in terms of the metric structure of the space. The proof uses the Hilbert invariant integral of the associated variational problem.

math.DG↗
arXiv2022

Roots of Polynomials and Umbilics of Surfaces

For certain polynomials we relate the number of roots inside the unit circle with the index of a non-degenerate isolated umbilic point on a real analytic surface in Euclidean 3-space. In particular, for $N>0$ we prove that for a certain ($N+2$)-real dimensional family of complex polynomials of degree $N$, the number of roots inside the unit circle is less than or equal to $1+N/2$. This bound is established as follows. From the polynomial we construct a convex real analytic surface containing an isolated umbilic point, such that the index of the umbilic point is determined by the number of roots of the polynomial that lie inside the unit circle. The bound on the number of roots then follows from Hamburger's bound on the index of an isolated umbilic point on a convex real analytic surface. The class of polynomials that arise are those with self-inversive second derivative. Thus the number of roots inside the unit circle is proven to be bounded for a polynomial with self-inversive second derivative.

math.DG↗
arXiv2021

Regularity and Continuity properties of the sub-Riemannian exponential map

We 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 Maslov index of Jacobi curves. We finally show how this implies that the exponential map of the three dimensional Heisenberg group is not injective in any neighbourhood of a conjugate vector.

math.DG↗
arXiv2020

Proof of the Toponogov Conjecture on Complete Surfaces

We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample. Corollaries of the main theorem include a Hawking-Penrose singularity-type theorem, as well as the proof of a conjecture of Milnor's from 1965 in the convex case.

math.DG↗
arXiv2018

Fredholm-Regularity of Holomorphic Discs in Plane Bundles over Compact Surfaces

We 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 a single complex point has $C^{2,α}$-close sections such that any (non-multiply covered) holomorphic disc with boundary in these sections are Fredholm regular. Fredholm regularity is also established when the complex surface is neutral Kähler, the action is both holomorphic and symplectic, and the section is Lagrangian with a single complex point.

math.AP↗
arXiv2017

Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows

In 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 rotationally symmetric sphere by a linear combination of its radii of curvature yields a Hopf sphere. When the coefficients of the flow have certain integer values, the fate of an initial sphere is entirely determined by the local geometry of its isolated umbilic points. A surprising variety of behaviours is uncovered: convergence to round spheres and non-round Hopf spheres, as well as divergence to infinity. The critical quantity is the rate of vanishing of the astigmatism - the difference of the radii of curvature - at the isolated umbilic points. It is proven that the size of this quantity versus the coefficient in the flow function determines the fate of the evolution. The geometric setting for the equation is Radius of Curvature space, viewed as a pair of hyperbolic/AdS half-planes joined along their boundary, the umbilic horizon. A rotationally symmetric sphere determines a parameterized curve in this plane with end-points on the umbilic horizon. The slope of the curve at the umbilic horizon is linked by the Codazzi-Mainardi equations to the rate of vanishing of astigmatism, and for generic initial conditions can be used to determine the outcome of the flow. The slope can jump during the flow, and a number of examples are given: instant jumps of the initial slope, as well as umbilic circles that contract to points in finite time and 'pop' the slope. Finally, we present soliton-like solutions: curves that evolve under linear flows by mutual hyperbolic/AdS isometries (dilation and translation) of Radius of Curvature space. A forthcoming paper will apply these geometric ideas to non-linear curvature flows.

math.DG↗
arXiv2016

A Global Version of a Classical Result of Joachimsthal

A 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 prove a global analogue of this result, as follows. Suppose that two closed convex surfaces intersect with constant angle along a curve that is not umbilic in either surface. We prove that the principal foliations of the two surfaces along the curve are either both orientable, or both non-orientable. We prove this by characterizing the constant angle intersection of two surfaces in Euclidean 3-space as the intersection of a surface and a hypersurface in the space of oriented lines. The surface is Lagrangian, while the hypersurface is null, with respect to the canonical neutral Kaehler structure. We establish a relationship between the principal directions of the two surfaces along the intersection curve in Euclidean space, which yields the result. This method of proof is motivated by topology and, in particular, the slice problem for curves in the boundary of a 4-manifold.

math.DG↗
arXiv2015

Parabolic Classical Curvature Flows

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we show decouples to highest order. We determine the conditions for this system to be parabolic and investigate the lower order terms. The zeroth order terms are shown to form a Hamiltonian system, which is therefore completely integrable. We find conditions on parabolic curvature flows that ensure boundedness of various geometric quantities and investigate some examples, including powers of mean curvature flow, Gauss curvature flow and mean radius of curvature flow, as well as the non-homogeneous Bloore flow. As a new tool we introduce the Radii of Curvature diagram of a surface and its canonical hyperbolic metric. The relationship between these and the properties of Weingarten surfaces are also discussed.

math.DG↗
arXiv2015

A Converging Lagrangian Curvature Flow in the Space of Oriented Lines

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the 3-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is determined by the initial surface. To prove this we show that the oriented normal lines, considered as a surface in the space of all oriented lines, evolve by a parabolic flow which preserves the Lagrangian condition. Moreover, this flow converges to a holomorphic Lagrangian section, which form the set of oriented lines through a point. The coordinates of this centre point are projections of the support function into the first non-zero eigenspace of the spherical Laplacian and are given by explicit integrals of initial surface data.

math.DG↗