Conical Singularities Research Articles

Optimal cone singularities for conformal flattening

Angle-preserving or conformal surface parameterization has proven to be a powerful tool across applications ranging from geometry processing, to digital manufacturing, to machine learning, yet conformal maps can still suffer from severe area distortion. Cone singularities provide a way to mitigate this distortion, but finding the best configuration of cones is notoriously difficult. This paper develops a strategy that is globally optimal in the sense that it minimizes total area distortion among all possible cone configurations (number, placement, and size) that have no more than a fixed total cone angle. A key insight is that, for the purpose of optimization, one should not work directly with curvature measures (which naturally represent cone configurations), but can instead apply Fenchel-Rockafellar duality to obtain a formulation involving only ordinary functions. The result is a convex optimization problem, which can be solved via a sequence of sparse linear systems easily built from the usual cotangent Laplacian. The method supports user-defined notions of importance, constraints on cone angles ( e.g. , positive, or within a given range), and sophisticated boundary conditions ( e.g. , convex, or polygonal). We compare our approach to previous techniques on a variety of challenging models, often achieving dramatically lower distortion, and demonstrating that global optimality leads to extreme robustness in the presence of noise or poor discretization.

Topological and geometric universal thermodynamics in conformal field theory

Universal thermal data in conformal field theory (CFT) offer a valuable means for characterizing and classifying criticality. With improved tensor network techniques, we investigate the universal thermodynamics on a nonorientable minimal surface, the crosscapped disk (or real projective plane, $\mathbb{RP}^2$). Through a cut-and-sew process, $\mathbb{RP}^2$ is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term $\frac{1}{2} \ln{k}$ related to nonorientable genus, with $k$ a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{c}{4} \ln{\beta}$, with $\beta$ the manifold size and $c$ the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.

Fuchsian Equations with Three Non-Apparent Singularities

We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients which maps the space of solutions of $H$ into the space of solutions of $E$. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations $E$ with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature $1$ on the punctured sphere with conic singularities, all but three of them having integer angles.

Hyperbolic vortices and Dirac fields in 2+1 dimensions

Starting from the geometrical interpretation of integrable vortices on two-dimensional hyperbolic space as conical singularities, we explain how this picture can be expressed in the language of Cartan connections, and how it can be lifted to the double cover of three-dimensional Anti-de Sitter space viewed as a trivial circle bundle over hyperbolic space. We show that vortex configurations on the double cover of AdS space give rise to solutions of the Dirac equation minimally coupled to the magnetic field of the vortex. After stereographic projection to (2+1)-dimensional Minkowski space we obtain, from each lifted hyperbolic vortex, a Dirac field and an abelian gauge field which solve a Lorentzian, (2+1)-dimensional version of the Seiberg–Witten equations.

Prescribed Gauss curvature problem on singular surfaces

We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $$\Sigma $$ admitting conical singularities of orders $$\alpha _i$$ ’s at points $$p_i$$ ’s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min–max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity $$\chi (\Sigma )+\sum _i \alpha _i$$ approaches a positive even integer, where $$\chi (\Sigma )$$ is the Euler characteristic of the surface $$\Sigma $$ .

Wave propagation on Euclidean surfaces with conical singularities. I: Geometric diffraction

We study wave diffraction on Euclidean surfaces with conic singularities (X, g) . We determine, for the first time, the precise microlocal structure of the wave at the intersection of the direct (or geometric) and diffracted fronts. Namely, we show that the wave kernel is a singular Fourier integral operator in a calculus associated to two intersecting Lagrangian submanifolds (corresponding to the two fronts), introduced originally byMelrose and Uhlmann [23]. We investigate the singularities of the trace of the half-wave group, Tr e^{-it\sqrt\Delta} , on (X, g) . We compute the leading-order singularity associated to periodic orbits with successive degenerate diffractions. This result extends the previous work of the third author [12] and the two-dimensional case of the work of the first author and Wunsch [10] as well as the seminal result of Duistermaat and Guillemin [7] in the smooth setting. In future work, we shall use these results to obtain inverse spectral results on Euclidean surfaces with conic singularities.

Drawing Cone Spherical Metrics via Strebel Differentials

Abstract Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. By using Strebel differentials as a bridge, we construct a new class of cone spherical metrics on compact Riemann surfaces by drawing on the surfaces some class of connected metric ribbon graphs.

Conic type Caffarelli\u2013Kohn\u2013Nirenberg inequality on manifold with conical singularity

In this paper, we consider a manifold with conical singularity and introduce weighted cone Sobolev spaces. We prove a conic type Caffarelli–Kohn–Nirenberg inequality and then apply this inequality to obtain a existence result of a nonlinear elliptic equation on weighted cone Sobolev space.

Hyperbolic ends with particles and grafting on singular surfaces

We prove that any 3-dimensional hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any 3-dimensional convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by “smooth grafting”.

Bounded $$H_\infty $$ H ∞ -calculus for cone differential operators

We prove that parameter-elliptic extensions of cone differential operators have a bounded $H_\infty$-calculus. Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities.

Elliptic complexes of first-order cone operators: ideal boundary conditions

The purpose of this paper is to provide a detailed description of the spaces that can be specified as L2 domains for the operators of a first order elliptic complex on a compact manifold with conical singularities. This entails an analysis of the nature of the minimal domain and of a complementary space in the maximal domain of each of the operators. The key technical result is the nondegeneracy of a certain pairing of cohomology classes associated with the indicial complex. It is further proved that the set of choices of domains leading to Hilbert complexes in the sense of Bruning and Lesch form a variety, as well as a theorem establishing a necessary and sufficient condition for the operator in a given degree to map its maximal domain into the minimal domain of the next operator.

Dihedral branched covers of four-manifolds

Given a closed oriented PL four-manifold X and a closed surface B embedded in X with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of X branched along B. For X simply-connected, we deduce a necessary condition on the intersection form of a simply-connected irregular dihedral branched cover of (X,B). When the singularities on B are two-bridge slice, we prove that the necessary condition on the intersection form of the cover is sharp. For X a simply-connected PL four-manifold with non-zero second Betti number, we construct infinite families of simply-connected PL manifolds which are irregular dihedral branched coverings of X. Given two four-manifolds X and Y whose intersection forms are odd, we obtain a necessary and sufficient condition for Y to be homeomorphic to an irregular dihedral p-fold cover of X, branched over a surface with a two-bridge slice singularity.

The Higher-Dimensional Chern\u2013Gauss\u2013Bonnet Formula for Singular Conformally Flat Manifolds

In a previous article, we generalised the classical four-dimensional Chern–Gauss–Bonnet formula to a class of manifolds with finitely many conformally flat ends and singular points, in particular obtaining the first such formula in a dimension higher than two which allows the underlying manifold to have isolated conical singularities. In the present article, we extend this result to all even dimensions nge 4 in the case of a class of conformally flat manifolds.

Asymptotically locally Euclidean/Kaluza–Klein stationary vacuum black holes in five dimensions

We produce new examples, both explicit and analytical, of bi-axisymmetric stationary vacuum black holes in five dimensions. A novel feature of these solutions is that they are asymptotically locally Euclidean, in which spatial cross-sections at infinity have lens space |$L(p,q)$| topology, or asymptotically Kaluza–Klein so that spatial cross-sections at infinity are topologically |$S^1\times S^2$|⁠. These are nondegenerate black holes of cohomogeneity 2, with any number of horizon components, where the horizon cross-section topology is any one of the three admissible types: |$S^3$|⁠, |$S^1\times S^2$|⁠, or |$L(p,q)$|⁠. Uniqueness of these solutions is also established. Our method is to solve the relevant harmonic map problem with prescribed singularities, having target symmetric space |$SL(3,\mathbb{R})/SO(3)$|⁠. In addition, we analyze the possibility of conical singularities and find a large family for which geometric regularity is guaranteed.

Does the black hole shadow probe the event horizon geometry?

There is an exciting prospect of obtaining the shadow of astrophysical black holes (BHs) in the near future with the Event Horizon Telescope. As a matter of principle, this justifies asking how much one can learn about the BH horizon itself from such a measurement. Since the shadow is determined by a set of special photon orbits, rather than horizon properties, it is possible that different horizon geometries yield similar shadows. One may then ask how sensitive is the shadow to details of the horizon geometry? As a case study, we consider the double Schwarzschild BH and analyse the impact on the lensing and shadows of the conical singularity that holds the two BHs in equilibrium -- herein taken to be a strut along the symmetry axis in between the two BHs. Whereas the conical singularity induces a discontinuity of the scattering angle of photons, clearly visible in the lensing patterns along the direction of the strut's location, it produces no observable effect on the shadows, whose edges remain everywhere smooth. The latter feature is illustrated by examples including both equal and unequal mass BHs. This smoothness contrasts with the intrinsic geometry of the (spatial sections of the) horizon of these BHs, which is not smooth, and provides a sharp example on how BH shadows are insensitive to some horizon geometry details. This observation, moreover, suggests that for the study of their shadows, this static double BH system may be an informative proxy for a dynamical binary.

Conical twist fields and null polygonal Wilson loops

Using an extension of the concept of twist field in QFT to space–time (external) symmetries, we study conical twist fields in two-dimensional integrable QFT. These create conical singularities of arbitrary excess angle. We show that, upon appropriate identification between the excess angle and the number of sheets, they have the same conformal dimension as branch-point twist fields commonly used to represent partition functions on Riemann surfaces, and that both fields have closely related form factors. However, we show that conical twist fields are truly different from branch-point twist fields. They generate different operator product expansions (short distance expansions) and form factor expansions (large distance expansions). In fact, we verify in free field theories, by re-summing form factors, that the conical twist fields operator product expansions are correctly reproduced. We propose that conical twist fields are the correct fields in order to understand null polygonal Wilson loops/gluon scattering amplitudes of planar maximally supersymmetric Yang–Mills theory.

A COMPACTNESS THEOREM FOR SURFACES WITH BOUNDED INTEGRAL CURVATURE

We prove a compactness theorem for metrics with bounded integral curvature on a fixed closed surface $\unicode[STIX]{x1D6F4}$. As a corollary we obtain a new convergence result for sequences of metrics with conical singularities, where an accumulation of singularities is allowed.

Vortices, Painlevé integrability and projective geometry

The first half of the thesis concerns Abelian vortices and Yang-Mills (YM) theory. It is proved that the 5 types of vortices recently proposed by Manton are symmetry reductions of (A)SDYM equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integrability results of such vortices can be derived. It is presented a natural definition of their kinetic energy and thus the metric of the moduli space was calculated by the Samols' localisation method. Then, a modified version of the Abelian-Higgs model is proposed in such a way that spontaneous symmetry breaking and the Bogomolny argument still hold. The Painleve test, when applied to its soliton equations, reveals a complete list of its integrable cases. The corresponding solutions are given in terms of third Painleve transcendents and can be interpreted as original vortices on surfaces with conical singularity. The last two chapters present the following results in projective differential geometry and Hamiltonians of hydrodynamic-type (HT) systems. It is shown that the projective structures defined by the Painleve equations are not metrisable unless either the corresponding equations admit first integrals quadratic in first derivatives or they define projectively flat structures. The corresponding first integrals can be derived from Killing vectors associated to the metrics that solve the metrisability problem. Secondly, it is given a complete set of necessary and sufficient conditions for an arbitrary affine connection in 2D to admit, locally, 0, 1, 2 or 3 Killing forms. These conditions are tensorial and simpler than the ones in previous literature. By defining suitable affine connections, it is shown that the problem of existence of Killing forms is equivalent to the conditions of the existence of Hamiltonian structures for HT systems of two components.

Multiple positive solutions for degenerate elliptic equations with singularity and critical cone Sobolev exponents

In this paper, we study the existence of multiple positive solutions for degenerate elliptic systems with singular and critical cone Sobolev exponents on singular manifolds. With the help of the variational method, we obtain a multiplicity result.

On flops and canonical metrics

This article is concerned with an observation for proving non-existence of canonical Kahler metrics. The idea is to use a rather explicit type of degeneration that applies in many situations. Namely, in a variation on a theme introduced by Ross-Thomas, we consider flops of the deformation to the normal cone. This yields a rather widely applicable notion of stability that is still completely explicit and readily computable, but with wider scope. We describe some applications, among them, a proof of one direction of the Calabi conjecture for asymptotically logarithmic del Pezzo surfaces. 1. Motivation and results A variety is slope stable in the sense of Ross-Thomas if, roughly, it is K-stable with respect to degenerations to the normal cone of its subvarieties. This notion has been studied extensively by a number of authors and has yielded many non-existence results for canonical metrics on projective Kahler manifolds. Our main purpose in this article is to introduce a slight variation on this theme by considering a somewhat more involved notion of stability that involves additional flops on the degeneration to the normal cone but that is still geometric and computable, and is partly inspired by the work of Li-Xu. This gives many new non-existence results, and most notably allows us to resolve one direction of the Calabi conjecture for asymptotically logarithmic del Pezzo surfaces. 1.1. Existence theorem for KEE metrics. Kahler-Einstein edge (KEE) metrics are a nat- ural generalization of Kahler-Einstein metrics: they are smooth metrics on the complement of a divisor, and have a conical singularity of angle 2πβ transverse to that 'complex edge'. When β = 1, of course, this is just an ordinary Kahler-Einstein metric, that extends smoothly across the divisor. One can think of the metric as being 'bent' at an angle 2πβ along the divisor. In the case of Riemann surfaces, KEE metrics are just the familiar constant curvature metrics with isolated cone singularities, that have been studied since the late 19th century, e.g., by Picard (25). A basic question, whose origins trace back to Tian's 1994 lectures in the setting of nonpositive curvature (32), extended in Donaldson's 2009 lectures to the setting of anticanonical divisors on Fano manifolds (10), and further extended in our previous work (5) (see also the survey (27, §8)), is the following: Problem 1.1. Under what analytic conditions on the triple (X,D,β) does a KEE metric exist on the Kahler manifold X bent at an angle 2πβ along the divisor D ⊂ X? This is partly motivated by Troyanov's solution in the Riemann surface case (36), and was settled by Jeffres-Mazzeo-Rubinstein (sufficient condition)and Darvas-Rubinstein (sufficient and necessary conditions) in higher dimensions for smooth D (17, 7). These results give an analytic criterion characterizing existence, once the cohomological condition