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Brendan Guilfoyle

Publications and source records attributed to Brendan Guilfoyle.

At least 37 records · Page 2Linked to original sources
arXiv2020

An Extension of Asgeirsson's Mean Value Theorem for Solutions of the ultra-hyperbolic Equation in Dimension Four

In 1937 Asgeirsson established a mean value property for solutions of the general ultra-hyperbolic equation in $2n$ variables. In the case of four variables, it states that the integrals of a solution over certain pairs of conjugate circles are the same. In this paper we extend this result to non-degenerate conjugate conics, which include the original case of conjugate circles and adds the new case of conjugate hyperbolae. The broader context of this result is the geometrization of Fritz John's 1938 analysis of the ultra-hyperbolic equation. Solutions of the equation arise as the compatibility for functions on line space to come from line integrals of functions in Euclidean 3-space. The introduction of the canonical neutral Kaehler metric on the space of oriented lines clarifies the relationship and broadens the paradigm to allow new insights. In particular, it is proven that a solution of the ultra-hyperbolic equation has the mean value property over any pair of curves that arise as the image of John's conjugate circles under a conformal map. These pairs of curves are then shown to be conjugate conics, which include circles and hyperbolae. John identified conjugate circles with the two rulings of a hyperboloid of 1-sheet. Conjugate hyperbolae are identified with the two rulings of either a piece of a hyperboloid of 1-sheet or a hyperbolic paraboloid.

math.AP↗
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↗
arXiv2018

A new geometric structure on tangent bundles

For a Riemannian manifold $(N,g)$, we construct a scalar flat metric $G$ in the tangent bundle $TN$. It is locally conformally flat if and only if either, $N$ is a 2-dimensional manifold or, $(N,g)$ is a real space form. It is also shown that $G$ is locally symmetric if and only if $g$ is locally symmetric. We then study submanifolds in $TN$ and, in particular, find the conditions for a curve to be geodesic. The conditions for a Lagrangian graph to be minimal or Hamiltonian minimal in the tangent bundle $T{\mathbb R}^n$ of the Euclidean real space ${\mathbb R}^n$ are studied. Finally, using the cross product in ${\mathbb R}^3$ we show that the space of oriented lines in ${\mathbb R}^3$ can be minimally isometrically embedded in $T{\mathbb R}^3$.

math.DG↗
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

The causal topology of neutral 4-manifolds with null boundary

This paper considers aspects of 4-manifold topology from the point of view of the null cone of a neutral metric, a point of view we call neutral causal topology. In particular, we construct and investigate neutral 4-manifolds with null boundaries that arise from canonical 3- and 4-dimensional settings. A null hypersurface is foliated by its normal and, in the neutral case, inherits a pair of totally null planes at each point. This paper focuses on these plane bundles in a number of classical settings The first construction is the conformal compactification of flat neutral 4-space into the 4-ball. The null foliation on the boundary in this case is the Hopf fibration on the 3-sphere and the totally null planes in the boundary are integrable. The metric on the 4-ball is a conformally flat, scalar-flat, positive Ricci curvature neutral metric. The second constructions are subsets of the 4-dimensional space of oriented geodesics in a 3-dimensional space-form, equipped with its canonical neutral metric. We consider all oriented geodesics tangent to a given embedded strictly convex 2-sphere. The third is a neutral geometric model for the intersection of two surfaces in a 4-manifold. The surfaces are the sets of oriented normal lines to two round spheres in Euclidean 3-space, which form Lagrangian surfaces in the 4-dimensional space of all oriented lines. The intersection of the boundaries of their normal neighbourhoods form tori that we prove are totally real and Lorentz if the spheres do not intersect. We conclude with possible topological applications of the three constructions, including neutral Kirby calculus, neutral knot invariants and neutral Casson handles, respectively.

math.DG↗
arXiv2016

Hopf hypersurfaces in spaces of oriented geodesics

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater than 2. For spherical and hyperbolic space forms, the oriented geodesic space admits a canonical Kaehler-Einstein and para-Kaehler-Einstein structure, respectively, so that a natural notion of a Hopf hypersurface exists. The particular hypersurfaces considered are formed by the oriented geodesics that are tangent to a given convex hypersurface in the underlying space form. We prove that a tangent hypersurface is Hopf in the space of oriented geodesics with respect to this canonical (para-)Kaehler structure iff the underlying convex hypersurface is totally umbilic and non-flat. In the case of 3 dimensional space forms, however, there exists a second canonical complex structure which can also be used to define Hopf hypersurfaces. We prove that in this dimension, the tangent hypersurface of a convex hypersurface in the space form is always Hopf with respect to this second complex structure.

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↗
arXiv2013

Marginally trapped surfaces in spaces of oriented geodesics

We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral Kaehler structures. We prove that every rank one surface in these four manifolds is marginally trapped. In the Euclidean case we show that Lagrangian rotationally symmetric sections are marginally trapped and construct an explicit family of marginally trapped Lagrangian tori. In the hyperbolic case we explore the relationship between marginally trapped and Weingarten surfaces, and construct examples of marginally trapped surfaces with various properties.

math.DG↗
arXiv2010

On the three-dimensional Blaschke-Lebesgue problem

The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n > 2. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n=3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant, and therefore are either spherical caps or pieces of tubes (canal surfaces).

math.DG↗
arXiv2009

On the Geometry of Spaces of Oriented Geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space (other than the octonion hyperbolic plane), 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 describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kaehler structure on L(M).

math.DG↗
arXiv2009

A characterization of Weingarten surfaces in hyperbolic 3-space

We study 2-dimensional submanifolds of the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ${\mathbb{H}}^3$ orthogonal to the geodesics of $Σ$. We prove that the induced metric on a Lagrangian surface in ${\mathbb{L}}({\mathbb{H}}^3)$ has zero Gauss curvature iff the orthogonal surfaces in ${\mathbb{H}}^3$ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in ${\mathbb{L}}({\mathbb{H}}^3)$ and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ${\mathbb{H}}^3$.

math.DG↗
arXiv2008

Totally Null Surfaces in Neutral Kaehler 4-Manifolds

We study the totally null surfaces of the neutral Kaehler 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 of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the $α$-planes are integrable and $α$-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The $β$-surfaces are less known and our interest is mainly in their description. In particular, we classify the $β$-surfaces of the neutral Kaehler metric on $TN$, the tangent bundle to a Riemannian 2-manifold $N$. These include the spaces of oriented geodesics in Euclidean and Lorentz 3-space, for which we show that the $β$-surfaces are affine tangent bundles to curves of constant geodesic curvature on $S^2$ and $H^2$, respectively. In addition, we construct the $β$-surfaces of the space of oriented geodesics of hyperbolic 3-space.

math.DG↗
arXiv2008

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Given an oriented Riemannian surface $(Σ, g)$, its tangent bundle $TΣ$ enjoys a natural pseudo-Kähler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic structure $\Om$. We give a local classification of those surfaces of $TΣ$ which are both Lagrangian with respect to $\Om$ and minimal with respect to $\G$. We first show that if $g$ is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in $\R^3$ or $\R^3_1$ induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in $T§^2$ or $T \H^2$ respectively. We relate the area of the congruence to a second-order functional $\mathcal{F}=\int \sqrt{H^2-K} dA$ on the original surface.

math.DG↗
arXiv2007

Lagrangian curves on spectral curves of monopoles

We study Lagrangian points on smooth holomorphic curves in T${\mathbb P}^1$ equipped with a natural neutral Kähler structure, and prove that they must form real curves. By virtue of the identification of T${\mathbb P}^1$ with the space ${\mathbb L}({\mathbb E}^3)$ of oriented affine lines in Euclidean 3-space ${\mathbb E}^3$, these Lagrangian curves give rise to ruled surfaces in ${\mathbb E}^3$, which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in ${\mathbb E}^3$, called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in ${\mathbb E}^3$ where the number of oriented lines in the complex curve $Σ$ that pass through the point is less than the degree of $Σ$. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.

math.DG↗
arXiv2007

On C$^2$-smooth Surfaces of Constant Width

A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along its normal lines. We also give a characterization of surfaces of constant width that have rational support function. Our techniques, which are complex differential geometric in nature, allow us to construct explicit smooth surfaces of constant width in ${\mathbb{E}}^3$, and their focal sets. They also allow for easy construction of tetrahedrally symmetric surfaces of constant width.

math.DG↗