Topological Disk Research Articles

Stripe Embedding: Efficient Maps with Exact Numeric Computation

We consider the fundamental problem of injectively mapping a surface mesh with disk topology onto a boundary constrained convex domain. We start from the basic observation that mapping a strip of triangles onto a rectangular shape always yields a valid embedding, if the vertices that bound the strip are sorted coherently along the sides of the rectangle. Based on this intuition, we propose a straightforward algorithm, called Stripe Embedding, that operates by decomposing the input mesh into a set of triangle strips and then embeds each strip into the target domain by means of linear interpolation between two previously embedded vertices. Thanks to its simplicity, Stripe Embedding is extremely efficient and permits to switch to an exact implementation without almost increasing its running times. Stripe Embedding is up to three orders of magnitude faster than the Tutte embedding for same numerical model and, even when implemented with costly rational numbers, it is faster than any floating point implementation of prior methods at any scale.

The q-Schwarzian and Liouville gravity

We present a new holographic duality between q-Schwarzian quantum mechanics and Liouville gravity. The q-Schwarzian is a one parameter deformation of the Schwarzian, which is dual to JT gravity and describes the low energy sector of SYK. We show that the q-Schwarzian in turn is dual to sinh dilaton gravity. This one parameter deformation of JT gravity can be rewritten as Liouville gravity. We match the thermodynamics and classical two point function between q-Schwarzian and Liouville gravity. We further prove the duality on the quantum level by rewriting sinh dilaton gravity as a topological gauge theory, and showing that the latter equals the q-Schwarzian. As the q-Schwarzian can be quantized exactly, this duality can be viewed as an exact solution of sinh dilaton gravity on the disk topology. For real q, this q-Schwarzian corresponds to double-scaled SYK and is dual to a sine dilaton gravity.

Piecewise Developable Modeling via Implicit Neural Deformation and Feature-Guided Cutting.

We propose a novel and automatic method to model shapes using a small set of discrete developable patches. Central to our approach is using implicit neural shape representation that makes our algorithm independent of tessellation and allows us to obtain the Gaussian curvature of each point analytically. With this powerful representation, we first deform the input shape to be an almost developable shape with clear and sparse salient feature curves. Then, we convert the deformed implicit field to a triangle mesh, which is further cut to disk topology along parts of the sparse feature curves. Finally, we achieve the resulting piecewise developable mesh by alternatingly optimizing discrete developability, enforcing manufacturability constraints, and merging patches. The feasibility and practicability of our method are demonstrated over various shapes. Compared to the state-of-the-art methods, our method achieves a better tradeoff between the number of developable patches and the approximation error.

Curvature varifolds with orthogonal boundary

AbstractWe consider the class of ‐dimensional surfaces in that intersect orthogonally along the boundary. A piece of an affine ‐plane in is called an orthogonal slice. We prove estimates for the area by the integral of the second fundamental form in three cases: first, when admits no orthogonal slices, second for if all orthogonal slices are topological disks, and finally, for all if the surfaces are confined to a neighborhood of . The orthogonality constraint has a weak formulation for curvature varifolds. We classify those varifolds of vanishing curvature. As an application, we prove for any the existence of an orthogonal 2‐varifold that minimizes the curvature in the integer rectifiable class.

Addendum to: M2-branes wrapped on a topological disk

Addendum to: M2-branes wrapped on a topological disk

On constant higher order mean curvature hypersurfaces in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$

Abstract We classify hypersurfaces with rotational symmetry and positive constant r-th mean curvature in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ . Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros–Rosenberg type theorem in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ : we show that compact connected hypersurfaces of constant r-th mean curvature embedded in H n × [ 0 , ∞ ) ${\mathbb{H}}^{n}{\times}\left[0,\infty \right)$ with boundary in the slice H n × { 0 } ${\mathbb{H}}^{n}{\times}\left\{0\right\}$ are topological disks under suitable assumptions.

Symmetry breaking and consistent truncations from M5-branes wrapping a disc

We construct new supersymmetric solutions corresponding to M5-branes wrapped on a topological disc by turning on additional scalars in the background. The presence of such scalar fields breaks one of the U(1) isometries of the internal space, explicitly realising the breaking by the Stückelberg mechanism observed previously. In addition, we construct a consistent truncation of maximal seven-dimensional gauged supergravity on the disc to five-dimensional Romans’ SU(2) × U(1) gauged supergravity, allowing us to construct a plethora of new supergravity solutions corresponding to more general states in the dual SCFTs as well as solutions corresponding to M5-branes wrapping four-dimensional orbifolds.

Hexagonalization of Fishnet integrals. Part II. Overlaps and multi-point correlators

This work presents the building-blocks of an integrability-based representation for multi-point Fishnet Feynman integrals with any number of loops. Such representation relies on the quantum separation of variables (SoV) of a non-compact spin-chain with symmetry SO(1, 5) explained in the first paper of this series. The building-blocks of the SoV representation are overlaps of the wave-functions of the spin-chain excitations inserted along the edges of a triangular tile of Fishnet lattice. The zoology of overlaps is analyzed along with various worked out instances in order to achieve compact formulae for the generic triangular tile. The procedure of assembling the tiles into a Fishnet integral is presented exhaustively. The present analysis describes multi-point correlators with disk topology in the bi-scalar limit of planar γ-deformed N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM theory, and it verifies some conjectural formulae for hexagonalisation of Fishnets CFTs present in the literature. The findings of this work are suitable of generalization to a wider class of Feynman diagrams.

Supersymmetric localization of (higher-spin) JT gravity: a bulk perspective

We study two-dimensional Jackiw-Teitelboim gravity on the disk topology by using a BF gauge theory in the presence of a boundary term. The system can be equivalently written in a supersymmetric way by introducing auxiliary gauginos and scalars with suitable boundary conditions on the hemisphere. We compute the exact partition function thanks to supersymmetric localization and we recover the result obtained from the Schwarzian theory by accurately identifying the physical scales. The calculation is then easily extended to the higher-spin generalization of Jackiw-Teitelboim gravity, finding perfect agreement with previous results. We argue that our procedure can also be applied to boundary-anchored Wilson lines correlators.

FZZ formula of boundary Liouville CFT via conformal welding

Liouville Conformal Field Theory (LCFT) on the disk describes the conformal factor of the quantum disk, which is the natural random surface in Liouville quantum gravity with disk topology. Fateev, Zamolodchikov and Zamolodchikov (2000) proposed an explicit expression, the so-called FZZ formula, for the one-point bulk structure constant for LCFT on the disk. In this paper we give a proof of the FZZ formula in the probabilistic framework of LCFT, which represents the first step towards rigorously solving boundary LCFT using conformal bootstrap. In contrast to previous works, our proof is based on conformal welding of quantum disks and the mating-of-trees theory for Liouville quantum gravity. As a byproduct of our proof, we also obtain the exact value of the variance for the Brownian motion in the mating-of-trees theory. Our paper is an essential part of an ongoing program proving integrability results for Schramm–Loewner evolutions, LCFT, and in the mating-of-trees theory.

The Poisson Bracket Invariant for Open Covers Consisting of Topological Disks on Surfaces

L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension $2$. In this note, instead of open covers consisting of displaceable sets in their work, we consider open covers constituting of topological disks and give a necessary and sufficient condition that Poisson bracket invariants of these covers are positive.

On monohedral tilings of a regular polygon

A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J Comb Theory Ser A 66:40–52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa et al. (Mediterr J Math 17:156, 2020) characterized the monohedral tilings of a circular disc by three topological discs. The aim of this note is to connect these two results by characterizing the monohedral tilings of any regular n-gon with at most three tiles for any n≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 5$$\\end{document}.

Towards a robust and portable pipeline for quad meshing: Topological initialization of injective integer grid maps

Integer Grid Maps (IGMs) are a class of mappings characterized by integer isolines that align up to unit translations and rotations of multiples of 90 degrees. They are widely used in the context of remeshing, to lay a quadrilateral grid onto the mapped surface. The presence of both discrete and continuous degrees of freedom makes the computation of IGMs extremely challenging. In particular, solving for all degrees of freedom altogether leads to a mixed-integer problem that is known to be NP-Hard. Such a problem can only be solved heuristically, occasionally failing to produce a valid quadrilateral mesh. In this paper we propose a simple topological construction that allows to reduce the problem of computing a valid IGM to the one of mapping a topological disk to a convex domain. This is a much easier problem to deal with, because it completely removes the integer constraints, permitting to obtain a provably injective parameterization that is guaranteed to incorporate all the correct integer transitions with a simple linear solve. Not only the proposed algorithm is easy to implement, but it is also independent from costly numerical solvers that are unavoidable in existing quadmeshing pipelines, preventing their exploitation in open source or low-budget projects. Despite provably correct, the so generated maps contain a considerable amount of geometric distortion and a poor quad connectivity, making this technique more suitable for a robust initialization rather than for the computation of an application-ready IGM. In the article we present the details of our construction, also analyzing its geometric and topological properties.

Constant mean curvature hypersurfaces in $\mathbb{H}^n\times\mathbb{R}$ with small planar boundary

We show that constant mean curvature hypersurfaces in {\mathbb H}^n\times\mathbb{R} , with small and pinched boundary contained in a horizontal slice P , are topological disks provided they are contained in one of the two halfspaces determined by P . This is the analogous in {\mathbb H}^n\times\mathbb{R} of a result in \mathbb{R}^3 by A. Ros and H. Rosenberg [J. Differential Geom. 44 (1996), 807–817].

Manifold-Constrained Geometric Optimization via Local Parameterizations.

Many geometric optimization problems contain manifold constraints that restrict the optimized vertices on some specified manifold surface. The constraints are highly nonlinear and non-convex, therefore existing methods usually suffer from a breach of condition or low optimization quality. In this article, we present a novel divide-and-conquer methodology for manifold-constrained geometric optimization problems. Central to our methodology is to use local parameterizations to decouple the optimization with hard constraints, which transforms nonlinear constraints into linear constraints. We decompose the input mesh into a set of developable or nearly-developable overlapping patches with disc topology, then flatten each patch into the planar domain with very low isometric distortion, optimize vertices with linear constraints and recover the patch. Finally, we project it onto the constrained manifold surface. We demonstrate the applicability and robustness of our methodology through a variety of geometric optimization tasks. Experimental results show that our method performs much better than existing methods.

Causal diamonds in ( 2+1 )-dimensional quantum gravity

We develop the reduced phase space quantization of causal diamonds in pure ($2+1$)-dimensional gravity with a nonpositive cosmological constant. The system is defined as the domain of dependence of a topological disc with fixed boundary metric. By solving the initial value constraints in a constant-mean-curvature time gauge and removing all the spatial gauge redundancy, we find that the phase space is the cotangent bundle of ${\text{Diff}}^{+}({S}^{1})/\mathrm{PSL}(2,\mathbb{R})$. To quantize this phase space we apply Isham's group-theoretic quantization scheme, with respect to a ${\mathrm{BMS}}_{3}$ group, and find that the quantum theory can be realized by wave functions on some coadjoint orbit of the Virasoro group, with labels in irreducible unitary representations of the corresponding little group. We find that the twist of the diamond boundary loop is quantized in integer or half-integer multiples of the ratio of the Planck length to the boundary length.

On p-Willmore disks with boundary energies

We consider an energy functional on surface immersions which includes contributions from both boundary and interior. Inspired by physical examples, the boundary is modeled as the center line of a generalized Kirchhoff elastic rod, while the interior term is arbitrarily dependent on the mean curvature and linearly dependent on the Gaussian curvature. We study equilibrium configurations for this energy in general among topological disks, as well as specifically for the class of examples known as p-Willmore energies.

Practical construction of globally injective parameterizations with positional constraints

We propose a novel method to compute globally injective parameterizations with arbitrary positional constraints on disk topology meshes. Central to this method is the use of a scaffold mesh that reduces the globally injective constraint to a locally flipfree condition. Hence, given an initial parameterized mesh containing flipped triangles and satisfying the positional constraints, we only need to remove the flips of a overall mesh consisting of the parameterized mesh and the scaffold mesh while always meeting positional constraints. To successfully apply this idea, we develop two key techniques. Firstly, an initialization method is used to generate a valid scaffold mesh and mitigate difficulties in eliminating flips. Secondly, edge-based remeshing is used to optimize the regularity of the scaffold mesh containing flips, thereby improving practical robustness. Compared to state-of-the-art methods, our method is much more robust. We demonstrate the capability and feasibility of our method on a large number of complex meshes.

Generalized Savitzky–Golay filter for smoothing triangular meshes

In this paper, the well-known Savitzky–Golay filter is generalized for three-dimensional triangular meshes. Savitzky–Golay filter is designed for smoothing noisy measurement signals, initially, a locally polynomial model is fitted to approximate the discrete measurement values. We will show, that the original idea can be naturally adapted for smoothing functions defined on an irregular two-dimensional triangular mesh. Primarily surfaces represented by three-dimensional triangular meshes are in the focus of this paper, which are by definition locally homeomorphic to the two-dimensional topological disk. As the generalized filter is meaningful on the plane, we will define local embeddings that send vertices and their neighborhoods to the disk. The points of the two-dimensional disk and the three-dimensional surface can be paired to each other using these embeddings, thereafter low-degree multidimensional polynomials can be fitted to the point pairs obtaining a similar continuous model, just like in the one-dimensional case. Finally, we show an application of our method for mesh smoothing, and a comparison to other mesh smoothing methods, e.g. the Laplacian smoothing and the Taubin smoothing.

Monohedral tilings of a convex disc with a smooth boundary

In this paper we give a complete description about normal monohedral tilings of a convex disc with smooth boundary where we have at most three topological discs as tiles. This result is a far-reaching generalization of the results of Kurusa, Lángi and Vígh [8]. Some further partial results are proved for non-normal tilings.