Boundary Components Research Articles

Recognition of objects through symplectic capacities

We prove that the generalized symplectic capacities recognize objects in symplectic categories whose objects are of the form (M,ω), such that M is a compact and 1-connected manifold, ω is an exact symplectic form on M, and there exists a boundary component of M with negative helicity. The set of generalized symplectic capacities is thus a complete invariant for such categories. This answers a question by Cieliebak, Hofer, Latschev, and Schlenk. It appears to be the first result concerning this question, except for recognition results for manifolds of dimension 2, ellipsoids, and polydiscs in R4. Strikingly, our result holds more generally for differential form categories. Recognition of objects is therefore not a symplectic phenomenon.We also prove a version of the result for normalized capacities.

Efficient estimation of boundary integrals for path-space differentiable rendering

Boundary integrals are unique to physics-based differentiable rendering and crucial for differentiating with respect to object geometry. Under the differential path integral framework---which has enabled the development of sophisticated differentiable rendering algorithms---the boundary components are themselves path integrals. Previously, although the mathematical formulation of boundary path integrals have been established, efficient estimation of these integrals remains challenging. In this paper, we introduce a new technique to efficiently estimate boundary path integrals. A key component of our technique is a primary-sample-space guiding step for importance sampling of boundary segments. Additionally, we show multiple importance sampling can be used to combine multiple guided samplings. Lastly, we introduce an optional edge sorting step to further improve the runtime performance. We evaluate the effectiveness of our method using several differentiable-rendering and inverse-rendering examples and provide comparisons with existing methods for reconstruction as well as gradient quality.

Asymptotic additivity of the Turaev–Viro invariants for a family of 3‐manifolds

In this paper, we show that the Turaev-Viro invariant volume conjecture posed by Chen and Yang is preserved under gluings of toroidal boundary components for a family of $3$-manifolds. In particular, we show that the asymptotics of the Turaev-Viro invariants are additive under certain gluings of elementary pieces arising from a construction of hyperbolic cusped $3$-manifolds due to Agol. The gluings of the elementary pieces are known to be additive with respect to the simplicial volume. This allows us to construct families of manifolds with an arbitrary number of hyperbolic pieces such that the resultant manifolds satisfy an extended version of the Turaev-Viro invariant volume conjecture.

Irreducible Metric Maps and Weil–Petersson Volumes

We consider maps on a surface of genus g with all vertices of degree at least three and positive real lengths assigned to the edges. In particular, we study the family of such metric maps with fixed genus g and fixed number n of faces with circumferences alpha _1,ldots ,alpha _n and a beta -irreducibility constraint, which roughly requires that all contractible cycles have length at least beta . Using recent results on the enumeration of discrete maps with an irreducibility constraint, we compute the volume V_{g,n}^{(beta )}(alpha _1,ldots ,alpha _n) of this family of maps that arises naturally from the Lebesgue measure on the edge lengths. It is shown to be a homogeneous polynomial in beta , alpha _1,ldots , alpha _n of degree 6g-6+2n and to satisfy string and dilaton equations. Surprisingly, for g=0,1 and beta =2pi the volume V_{g,n}^{(2pi )} is identical, up to powers of two, to the Weil–Petersson volume V_{g,n}^{mathrm {WP}} of hyperbolic surfaces of genus g and n geodesic boundary components of length L_i = sqrt{alpha _i^2 - 4pi ^2}, i=1,ldots ,n. For genus gge 2 the identity between the volumes fails, but we provide explicit generating functions for both types of volumes, demonstrating that they are closely related. Finally we discuss the possibility of bijective interpretations via hyperbolic polyhedra.

Quantitative Comparison of a Hierarchy of Commonly Used Planetary Climate Energy Balance Models

The objective of this paper is to examine the effect of the various simplifications inherent in commonly-used planetary energy balance climate models (EBMs). Specifically, we look at the zero-dimensional (0-d) radiative equilibrium model, the 1-d radiative equilibrium model, and the 1-d radiative-convective equilibrium (RCE) model. Each of these models make fundamental steps towards a well-represented Earth system model, and make different simplifications and assumptions in the process. We seek to evaluate the effects these assumptions have on key thermal quantities of the system (OLR / outgoing longwave radiation, surface temperature, etc.). These evaluations lead us to identify contexts for each model wherein it remains a valid option to accurately replicate a system. The 0-d model fails to account for the greenhouse effect’s impact on energetics, thus predicting an erroneously-low surface temperature and low OLR. It instead requires an emissivity coefficient ~ 0.619 to balance OLR and temperature and model the Earth system. The 1-d radiative equilibrium model is a significant improvement on its predecessor, creating a stratospheric thermal profile reasonably similar to that of the Earth. The strong low-altitude temperature lapse rate and convective instability near the surface, however, slightly diminishes the validity of the low-level thermal profile – a drawback the RCE model appears to resolve with the addition of convective and boundary layer components. We conclude that the 0-d radiative equilibrium is best suited to isothermal atmospheres, the 1-d radiative equilibrium model to non-isothermal atmospheres where convection is suppressed, and the 1-d RCE model to convectively-active atmospheres.

Spun normal surfaces in 3-manifolds III: Boundary slopes

Spun normal surfaces are a useful way of representing proper essential surfaces, using ideal triangulations for 3-manifolds with tori and Klein bottle boundaries. In this paper, we consider spinning essential surfaces in an irreducible P 2 P^2 -irreducible, anannular, atoroidal 3-manifold with tori and Klein bottle boundary components. We can assume that such a 3-manifold is equipped with an ideal 1-efficient triangulation. In particular, we prove that for a given choice of a set of boundary slopes for a proper essential surface, there is a set of essential vertex solutions for the projective solution space at these boundary slopes, answering a question of Dunfield and Garoufalidis [Trans. Amer. Math. Soc. 364 (2012), pp. 6109–6137], so long as the slopes are not of a fiber of a bundle structure.

A new class of discrete conformal structures on surfaces with boundary

We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpretations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using variational principles, which is a complement of Guo–Luo’s rigidity of the discrete conformal structures in Guo and Luo (Geom Topol 13(3):1265–1312, 2009) and Guo’s rigidity of vertex scaling in Guo (Commun Contemp Math 13(5):827–842, 2011) on surfaces with boundary. As a byproduct, new results on the convexity of the volume of generalized hyperbolic pyramids with right-angled hyperbolic hexagonal bases are obtained. Motivated by Chow–Luo’s combinatorial Ricci flow and Luo’s combinatorial Yamabe flow on closed surfaces, we further introduce combinatorial Ricci flow and combinatorial Calabi flows to deform the new discrete conformal structures on surfaces with boundary. The basic properties of these combinatorial curvature flows are established. These combinatorial curvature flows provide effective algorithms for constructing hyperbolic metrics on surfaces with totally geodesic boundary components of prescribed lengths.

Diagonal complexes for surfaces of finite type and surfaces with involution

Two constructions are studied that are inspired by the ideas of a recent paper by the authors. — The diagonal complex D \mathcal {D} and its barycentric subdivision B D \mathcal {BD} related to an oriented surface of finite type F F equipped with a number of labeled marked points. This time, unlike the paper mentioned above, boundary components without marked points are allowed, called holes. — The symmetric diagonal complex D inv \mathcal {D}^{\operatorname {inv}} and its barycentric subdivision B D inv \mathcal {BD}^{\operatorname {inv}} related to a symmetric (=with an involution) oriented surface F F equipped with a number of (symmetrically placed) labeled marked points. The symmetric complex is shown to be homotopy equivalent to the complex of a surface obtained by “taking a half” of the initial symmetric surface.

Curves on Non-Orientable Surfaces and Crosscap Transpositions

Let Ng,n be a non-orientable surface of genus g with n punctures and one boundary component. In this paper, we describe multicurves in Ng,n making use of generalized Dynnikov coordinates, and give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in Ng,n in terms of generalized Dynnikov coordinates. This provides a way to solve on non-orientable surfaces various dynamical and combinatorial problems that arise in the study of mapping class groups and Thurston’s theory of surface homeomorphisms, which were solved only on orientable surfaces before.

Disorder averaging and its UV discontents

We present a stringy realization of quantum field theory ensembles in $D\ensuremath{\le}4$ spacetime dimensions, thus realizing a disorder averaging over coupling constants. When each member of the ensemble is a conformal field theory with a standard semiclassical holographic dual of the same radius, the resulting bulk can be interpreted as a single asymptotically anti--de Sitter space geometry with a distribution of boundary components joined by wormhole configurations, as dictated by the Hartle--Hawking wave function. This provides a UV completion of a recent proposal by Marolf and Maxfield that there is a high-dimensional Hilbert space for baby universes, but one that is compatible with the proposed swampland constraints of McNamara and Vafa. This is possible because our construction is really an approximation that breaks down both at short distances, but also at low energies for objects with a large number of microstates. The construction thus provides an explicit set of counterexamples to various claims in the literature that holographic and effective field theory considerations can be reliably developed without reference to any UV completion.

A high-genus asymptotic expansion of Weil–Petersson volume polynomials

The object under consideration in this article is the total volume Vg,n(x1, …, xn) of the moduli space of hyperbolic surfaces of genus g with n boundary components of lengths x1, …, xn, for the Weil–Petersson volume form. We prove the existence of an asymptotic expansion of the quantity Vg,n(x1, …, xn) in terms of negative powers of the genus g, true for fixed n and any x1, …, xn ≥ 0. The first term of this expansion appears in the work of Mirzakhani and Petri [Comment. Math. Helvetici 94, 869–889 (2019)], and we compute the second term explicitly. The main tool used in the proof is Mirzakhani’s topological recursion formula, for which we provide a comprehensive introduction.

Interphase boundary, grain boundary, and surface diffusion in Al2O3-GdAlO3 composites determined from bicrystal coble creep experiments

Small-scale in situ transmission electron microscopy-based Coble creep experiments were performed on Al2O3-GdAlO3 composites using localized laser heating. A primary goal of the work was to isolate the Al2O3-GdAlO3 interphase boundary diffusivity in order to understand how it contributes to the average properties of the composite. The diffusivities of the grain boundaries (GB) of GdAlO3 (GAP) and Al2O3 were measured to be DGb,GAP=15±14exp−580,000±62,000Jmol−1RTm2s−1, and DGB,Al2O3=25±21exp−542,000±30,000Jmol−1RTm2s−1, respectively. The average interphase boundary (IPB) diffusivity exceeds that of the grain boundaries and was measured to be DIPB=15±14exp−559,000±117,000Jmol−1RTm2s−1. Capillary smoothing experiments after creep were used to determine surface (S) diffusivities, DS,Al2O3=3.2x103±2.7x103exp−539,000±26,000Jmol−1RTm2s−1 and DS,GAP=4.6x104±7.7x10−6exp−625,000±82,000Jmol−1RTm2s−1. This works demonstrates the feasibility of small-scale Coble creep experiments to directly measure individual components of grain boundary and IPB diffusivities in composites.

CONNECTEDNESS OF THE CUT SYSTEM COMPLEX ON NONORIENTABLE SURFACES

Let N be a compact, connected, nonorientable surface of genus g with n boundary components. In this note, we show that the cut system complex of N is connected for g < 4 and disconnected for g ≥ 4. We then define a related complex and show that it is connected for g ≥ 4.

A note on arclength null quadrature domains

We prove the existence of a roof function for arclength null quadrature domains having finitely many boundary components. This bridges a gap toward classification of arclength null quadrature domains by removing an a priori assumption from previous classification results.

Intersection bodies of polytopes

We investigate the intersection body of a convex polytope using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is always a semialgebraic set and provide an algorithm for its computation. Moreover, we compute the irreducible components of the algebraic boundary and provide an upper bound for the degree of these components.

Exhausting curve complexes by finite rigid sets on nonorientable surfaces

Let [Formula: see text] be a compact, connected, nonorientable surface of genus [Formula: see text] with [Formula: see text] boundary components. Let [Formula: see text] be the curve complex of [Formula: see text]. We prove that if [Formula: see text] or [Formula: see text], then there is an exhaustion of [Formula: see text] by a sequence of finite rigid sets. This improves the author’s result on exhaustion of [Formula: see text] by a sequence of finite superrigid sets.

Multiple curves on punctured orientable surfaces

We describe each multiple curve on an orientable surface of genus-1, with n punctures and one boundary component by using this multiple curve?s geometric intersection numbers with the embedded curves in this surface.

Double-ended fast protection system of LCC-VSC-MTDC independent on boundary component

Double-ended fast protection system of LCC-VSC-MTDC independent on boundary component

Bounds for the Morse index of free boundary minimal surfaces

Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of Fraser-Li, we conclude that the area index of a free-boundary minimal surface in a convex domain of Euclidean three-space, is bounded from above by a linear function of its genus and its number of boundary components. We also prove index bounds for submanifolds of higher dimension.

Diffstrata—A Sage Package for Calculations in the Tautological Ring of the Moduli Space of Abelian Differentials

The boundary of the multi-scale differential compactification of strata of abelian differentials admits an explicit combinatorial description. However, even for low-dimensional strata, the complexity of the boundary requires use of a computer. We give a description of the algorithms implemented in the SageMath package diffstrata to enumerate the boundary components and perform intersection theory on this space. In particular, the package can compute the Euler characteristic of strata using the methods developed by the same authors.