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Brendan Guilfoyle

Publications and source records attributed to Brendan Guilfoyle.

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

Isotropic submanifolds of $T\mathbb{S}^n$ and their focal sets

Families of oriented lines in $\mathbb{R}^{n+1}$ are studied via their identification with submanifolds of $T\mathbb{S}^n$. In particular, families of oriented lines which are orthogonal to submanifolds in $\mathbb{R}^{n+1}$ are shown to characterise those which are isotropic with respect to the canonical sympleptic structure on $T\mathbb{S}^n$. Families of lines that are tangent to a $k$-dimensional submanifold of $\mathbb{R}^{n+1}$ are then studied. For such families, isotropy is shown to be equivalent to the generating vector field being geodesic and hypersurface-orthogonal on the submanifold. The focal set in $\mathbb{R}^{n+1}$ of a family of lines is introduced, extending the classical definition for families normal to hypersurfaces, to general families of lines of arbitrary codimension. A formula is derived that expresses certain sectional curvatures of the focal set in terms of the signed distances between corresponding focal points. We then solve an inverse problem for the focal sets of hypersurfaces and show certain sectional and Ricci curvatures of the focal set are determined by the differences between the hypersurface's radii of curvature. This generalises a Theorem of Bianchi from 1874 - namely that surfaces in $\mathbb{R}^3$ of constant astigmatism have pseudo-spherical focal sets.

math.DG↗
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↗
arXiv2024

Minimal surfaces in the Riemannian product of surfaces

Minimal surfaces in the Riemannian product of surfaces of constant curvature have been considered recently, particularly as these products arise as spaces of oriented geodesics of 3-dimensional space-forms. This papers considers more general Riemannian products of surfaces and explores geometric and topological restrictions that arise for minimal surfaces. We show that generically, a totally geodesic surface in a Riemannian product is locally either a slice or a product of geodesics. If the Gauss curvatures of the factors are negative, it is proven that there are no minimal 2-spheres, while minimal 2-tori are Lagrangian with respect to both product symplectic structures. If the surfaces have non-zero bounded curvatures, we establish a sharp lower bound on the area of minimal 2-spheres and explore the properties of the Gauss and normal curvatures of general compact minimal surfaces.

math.DG↗
arXiv2023

A Note on Umbilic Points at Infinity

In this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed by the authors in a previous paper. A geometric interpretation for our definition is that an umbilic point at infinity occurs when the tangent to the level set at infinity is also an asymptotic direction at infinity. We prove that a homogeneous polynomial graph must have an umbilic point, albeit at infinity.

math.DG↗
arXiv2023

From CT scans to 4-manifold topology

In this survey paper the ultrahyperbolic equation in dimension four is discussed from a geometric, analytic and topological point of view. The geometry centres on the canonical neutral metric on the space of oriented geodesics of 3-dimensional space-forms, the analysis discusses a mean value theorem for solutions of the equation and presents a new solution of the Cauchy problem over a certain family of null hypersurfaces, while the topology relates to generalizations of codimension two foliations of 4-manifolds.

math.DG↗
arXiv2023

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↗
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↗
arXiv2022

Properties and Transformations of Weingarten Surfaces

The Weingarten relations satisfied by rotationally symmetric surfaces in Euclidean 3-space E3 are considered from three points of view: restrictions on the slope of the relation at umbilic points, the action of SL2(R) as fractional linear transformations on the space of curvatures, and variational formulations for the relations. With regard to the first, we obtain bounds on the slope of a Weingarten relation in terms of the fall off of the radii of curvature at an umbilic point. This generalizes recent work by a number of authors. For the second, we show that the action descends from curvature space to E3 and splits into three natural geometric actions. This is applied to a class of Weingarten surfaces, called semi-quadratic, on which the action is shown to be transitive. Finally, a natural Lagrangian formulation is given for certain types of Weingarten relations and stability established.

math.DG↗
arXiv2022

A Conformal Mean Value Theorem for Solutions of the Ultrahyperbolic Equation

Asgeirsson's theorem establishes a mean value property for solutions of the ultrahyperbolic equation. 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, the invariance of the four dimensional ultrahyperbolic equation under conformal maps of the pseudo-Euclidean space of signature 2+2 is used to get the most general version of the mean value theorem. The name non-degenerate conjugate conics is used for the most general pairs of curves over which solutions of the ultrahyperbolic equation enjoy the mean value property. These are proven to be pairs of conic sections, so that, in addition to conjugate circles which were known to exist, the picture is completed by finding mean value theorems over conjugate hyperbolae, conjugate parabolae, and line-empty pairs. In addition, Fritz John established a link between conjugate circles and the two rulings of a hyperboloid of revolution. The line incidence property of doubly ruled surfaces is used to prove a one-to-one correspondence between non-degenerate conjugate conics and pairs of rulings of doubly ruled surfaces in Euclidean 3-space.

math.DG↗
arXiv2022

On the Convergence of Non-Integer Linear Hopf Flow

The evolution of a rotationally symmetric surface by a linear combination of its radii of curvature equation is considered. It is known that if the coefficients form certain integer ratios the flow is smooth and can be integrated explicitly. In this paper the non-integer case is considered for certain values of the coefficients and with mild analytic restrictions on the initial surface. We prove that if the focal points at the north and south poles on the initial surface coincide, the flow converges to a round sphere. Otherwise the flow converges to a non-round Hopf sphere. Conditions on the fall-off of the astigmatism at the poles of the initial surface are also given that ensure the convergence of the flow. The proof uses the spectral theory of singular Sturm-Liouville operators to construct an eigenbasis for an appropriate space in which the evolution is shown to converge.

math.DG↗
arXiv2022

A Uniqueness Theorem for Incompressible Fluid Flows with Straight Streamlines

It is proven that the only incompressible Euler fluid flows with fixed straight streamlines are those generated by the normal lines to a round sphere, a circular cylinder or a flat plane, the fluid flow being that of a point source, a line source or a plane source at infinity, respectively. The proof uses the local differential geometry of oriented line congruences to integrate the Euler equations explicitly.

math.AP↗
arXiv2021

Almost Paracomplex Structures on 4-Manifolds

Reflection in a line in Euclidean 3-space defines an almost paracomplex structure on the space of all oriented lines, isometric with respect to the canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of oriented geodesics of any real 3-dimensional space form admits both isometric and anti-isometric paracomplex structures. This paper considers the existence or otherwise of isometric and anti-isometric almost paracomplex structures $j$ on a pseudo-Riemannian 4-manifold $(M,g)$, such that $j$ is parallel with respect to the Levi-Civita connection of $g$. It is shown that if an isometric or anti-isometric almost paracomplex structure on a conformally flat manifold is parallel, then the scalar curvature of the metric must be zero. In addition, it is found that $j$ is parallel iff the eigenplanes are tangent to a pair of mutually orthogonal foliations by totally geodesic surfaces. The composition of a Riemannian metric with an isometric almost paracomplex structure $j$ yields a neutral metric $g'$. It is proven that if $j$ is parallel, then $g$ is Einstein iff $g'$ is conformally flat and scalar flat. The vanishing of the Hirzebruch signature is found to be a necessary topological condition for a closed 4-manifold to admit an Einstein metric with a parallel isometric paracomplex structure. Thus, while the K3 manifold admits an Einstein metric with an isometric paracomplex structure, it cannot be parallel. The same holds true for certain connected sums of complex projective 2-space and its conjugate.

math.DG↗
arXiv2021

Umbilic Points on the Finite and Infinite Parts of Certain Algebraic Surfaces

The global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function $f$ on the plane are studied. We provide a Poincaré-Hopf type formula where the sum over all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of $f$. Moreover, we study the projective extension of these fields and prove, under generic conditions, that every umbilic point at infinity of these extensions is isolated, has index equal to 1/2 and its topological type is a Lemon.

math.DG↗
arXiv2021

On Isolated Umbilic Points

Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3-space. In particular, Riemannian metrics with a smooth strictly convex 2-sphere containing a single umbilic point are constructed explicitly, in contradiction with any direct extension of Caratheodory's conjecture. Additionally, a Riemannian metric with an embedded surface containing an isolated umbilic point of any index is presented, violating Hamburger's umbilic index bound. In both cases, it is shown that the metric can be made arbitrarily close to the flat metric. A short video explaining the motivation and results of this paper can be found at the following link: https://youtu.be/Wjja4PcMtxc

math.DG↗
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↗
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↗