Generalizes complex manifolds to manifolds with corners and generalized corners.
problem Tackles the extension of complex structures to manifolds with corners and generalized corners.
method Uses complex structures on the b-tangent bundle and proves a formal Newlander-Nirenberg type theorem.
result Proves that along each corner stratum, the b-complex structure agrees with a standard model to infinite order.
In conventional Differential Geometry one studies manifolds, locally modelled on Rn, manifolds with boundary, locally modelled on [0,∞)×Rn−1, and manifolds with corners, locally modelled on [0,∞)k×Rn−k. They form categories ${\bf Man}\subset{\bf Man^b}\sub…
Manifolds with boundary and with corners form categories Man⊂Manb⊂Manc. A manifold with corners X has two notions of tangent bundle: the tangent bundle TX, and the b-tangent bundle bTX. The usual definition of smooth structure uses TX, as f:X→R is defined to be …
Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 c…
Unified framework for gravity on manifolds with corners.
problem Unified description of gravity on manifolds with corners.
method Derivation of corner Poisson structure, reduction procedure, equivalent affine Poisson description.
result Unified framework for bulk, boundary, and corner structures of Palatini-Cartan gravity.
Unified framework for gravity on manifolds with corners derived.
problem Unified description of gravity on manifolds with corners.
method Derivation of corner Poisson structure and reduction procedure.
result Unified framework for bulk, boundary, and corner structures of Palatini-Cartan gravity.
New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.
problem Defining curvature measures for 4D manifolds with corners.
method Defined two new extrinsic curvature quantities, one conformal invariant, and a new conformally invariant operator.
result Gauss-Bonnet theorem reformulated in terms of new curvature measures.
Defines products for fibered corners manifolds, generalizing resolutions.
problem Resolving fibered corners manifolds.
method Introduces a category of fibered corners manifolds with products and transverse fiber products, defining the 'ordered product' for wedge metrics.
result The 'ordered product' is a natural product for wedge metrics.
Extends 4D cornered skein theory to surfaces, proving gluing formulas.
problem Formulating gluing formulas for 4-manifolds with corners and boundaries.
method Develops a categorical framework and introduces bicategories for closed surfaces.
result Proves gluing formulas for categories associated with 3-manifolds with boundary.
This paper proves a normal form for cornered asymptotically hyperbolic metrics.
problem Cornered asymptotically hyperbolic metrics and their geometric properties.
method Proves a Cartan-Hadamard type theorem for the normal exponential map and constructs a normal form.
result Normal form for cornered asymptotically hyperbolic metrics.
Extends algebraic geometry to include spaces with corners.
problem Generalizing manifolds with corners.
method Defines and studies C∞-rings and schemes with corners. result New categories of C∞-rings and schemes with corners. Paper proves rigidity for Penrose inequality on 3-manifolds with corners.
problem Proving rigidity for Penrose inequality on specific 3-manifolds.
method Analyzes asymptotically flat manifolds with nonnegative scalar curvature and corners.
result Proves rigidity for equality cases of Penrose inequality.
Generalizes differentiation under integral sign to submanifolds with corners.
problem Closing a gap in mathematical literature for evolving submanifolds with corners.
method Proves generalizations of the Reynolds Transport Theorem for submanifolds with corners.
result Provides a unified treatment of integral theorems for unbounded cases.
We construct a smooth Lie group structure on the group of real analytic diffeomorphisms of a compact analytic manifold with corners. This generalises the known analogous results in the situation where the real analytic manifold has no corners. Additionally our approach uses a different construction.
Calculates the index of a geometric Dirac operator on manifolds with corners using glueing and Lie groupoid.
problem Calculating the Fredholm index of a geometric Dirac operator with mixed boundary conditions.
method Introduces a glueing construction and Lie groupoid to describe the Dirac operator. Uses a heat kernel method with rescaling to derive an index formula.
result Derives a general index formula of the Atiyah-Singer type.
The paper studies geometric obstructions for Fredholm conditions on manifolds with corners.
problem Geometric obstructions for Fredholm conditions on manifolds with corners.
method Homology theory called conormal homology, Euler characteristic computation.
result Explicit computation of K-theory groups for b-compact operators.
The paper solves obstructions to the Fredholm perturbation property for manifolds with corners.
problem Obstructions to the Fredholm perturbation property for compact connected manifolds with corners.
method Introduces a topological space whose singular cohomology is canonically isomorphic to conormal homology and whose K-theory is naturally isomorphic to the K-theory groups of the algebra K_b(X).
result Provides a rational isomorphism between K-theory groups and periodic conormal homology groups, solving obstructions to the Fredholm perturbation property.
Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
Defines s-manifolds and s-manifolds with corners for symplectic applications.
problem Creating a framework for singular manifolds with useful properties.
method Introducing categories of stratified manifolds and manifolds with corners.
result Fundamental classes and transverse fibre products in s-manifolds.
In this paper we present another notion of a smooth manifold with corners and relate it to the commonly used concept in the literature. Afterwards we introduce complex manifolds with corners and show that if M is a compact (respectively complex) manifold with corners and K is a smooth (respectively complex) Lie gro…
The paper constructs submanifolds with corners in Delzant polytopes from affine subspaces.
problem Understanding submanifolds with corners in Delzant polytopes.
method Constructing submanifolds with corners in Delzant polytopes from affine subspaces.
result Conditions for submanifolds with corners are equivalent to those for torus-equivariantly embedded toric manifolds.
M-theory can be defined on closed manifolds as well as on manifolds with boundary. As an extension, we show that manifolds with corners appear naturally in M-theory. We illustrate this with four situations: The lift to bounding twelve dimensions of M-theory on Anti de Sitter spaces, ten-dimensional heterotic string the…
Manifolds without boundary, and manifolds with boundary, are universally known in Differential Geometry, but manifolds with corners (locally modelled on [0,\infty)^k x R^{n-k}) have received comparatively little attention. The basic definitions in the subject are not agreed upon, there are several inequivalent definiti…
We introduce the notions of the caustic-equivalence and the weak caustic-equivalence relations of reticular Lagrangian maps in order to give a generic classification of caustics on a corner. We give the figures of all generic caustics on a corner in a smooth manifold of dimension 2 and 3.
Extends corner structure study to general case, constructs normal Trans-Sasakian structures.
problem Extending corner structure study to general case without conditions.
method Extends corner structure to general case, constructs Trans-Sasakian structures from non-normal corner structures.
result Constructs normal Trans-Sasakian structures from non-normal corner structures.
New framework for logarithmically divergent integrals on manifolds with corners.
problem Logarithmically divergent integrals on manifolds with corners.
method Introduces new geometric framework and morphisms in logarithmic geometry.
result Functorial characterization of regularized integration.
The study of Einstein metrics with corners and boundaries.
problem Formal study of Einstein metrics with corners and boundaries.
method Generalization and extension of previous work; formal expansion and existence demonstration.
result Existence of Einstein metrics up to a certain order in a cornered asymptotically hyperbolic normal form.
This paper is concerned with pseudodifferential calculus on manifolds with fibred corners. Following work of Connes, Monthubert, Skandalis and Androulidakis, we associate to every manifold with fibred corners a longitudinally smooth groupoid which algebraic and differential structure is explicitely described. This grou…
Smoothly attaches manifolds with controlled curvature.
problem Attaching manifolds with corners and controlling curvature.
method Desingularization theorem to smoothly attach manifolds with controlled scalar and mean curvature.
result Smoothly attaches manifolds with controlled scalar curvature and mean curvature of boundary.
Develops a facewise formulation of AKSZ construction on manifolds with ordinary corners.
problem Formulating AKSZ construction on manifolds with ordinary corners.
method Explicit formal mapping-space hypothesis, facewise formulation, organization over face poset, Hamiltonian defect, face incidence complex.
result Factorially normalized facewise transgression is a cochain map, closed target forms transgress to cocycles, and twice-iterated defect vanishes.
Introduces generalized products for pseudodifferential operators on manifolds with corners.
problem Developing a new algebraic structure for pseudodifferential operators.
method Introduces generalized products and shows their implications for pseudodifferential operators.
result Generalized products imply the existence of an algebra of pseudodifferential operators.
We construct a functor from the category of manifolds with generalized corners to the category of complexes of toric monoids, and for every `refinement' of the complex associated to a manifold, we show there is a unique `blow-up', i.e., a new manifold mapping to the original one, which satisfies a universal property an…
We show that there is a fully faithful embedding of the category of manifolds with corners into the Cahiers topos, one of the premier models for Synthetic Differential Geometry. This embedding is shown to have a number of nice properties, such as preservation of open covers and transverse fibre products. We develop a t…
Functorial compactification of vector spaces defined as manifolds with corners.
problem Compactification of vector spaces functorial under linear maps.
method Definition of manifolds with corners and b-maps, application of iterated blow-up theory.
result Criterion for lifted maps to be b-fibrations, identification of restrictions to boundary hypersurfaces.
We prove that in a class of non-equiregular sub-Riemannian manifolds corners are not length minimizing. This extends the results [4]. As an application of our main result we complete and simplify the analysis in [6], showing that in a 4-dimensional sub-Riemannian structure suggested by Agrachev and Gauthier all length-…
Compactifies semi-simple Lie groups to manifolds with corners.
problem Compactification of semi-simple Lie groups.
method hd-compactification to a manifold with corners, 1-1 correspondence with conjugacy classes of parabolic subgroups.
result Harish-Chandra's Schwartz space identified with conormal functions of rapid-logarithmic decay.
We introduce the notion of reticular Legendrian unfoldings in order to investigate stabilities of bifurcations of wavefronts generated by a hypersurface germ with a boundary, a corner, or an r-corner in a smooth n dimensional manifold. We define several stabilities of reticular Legendrian unfoldings and prove that they…
New proof of Riemannian Penrose Inequality for manifolds with corners
problem Riemannian Penrose Inequality for asymptotically flat manifolds with corners
method Unified argument based on approximate monotonicity
result Positive Mass Theorem and Riemannian Penrose Inequality
Generalizes moment-angle manifolds to arbitrary nice manifolds with corners.
problem Computing cohomology groups and rings for moment-angle manifolds.
method Stable decomposition, rim-cubicalization, partial diagonal maps, polyhedral product.
result Derived formulas for integral cohomology groups and rings of moment-angle manifolds.
Moser's theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Moser's theorem on simplices. The proof is based on Banyag…
Classifies actions of tori on manifolds up to diffeomorphisms.
problem Classifying actions of tori on manifolds up to diffeomorphisms.
method Using triples (Q, λ, c) to classify actions, where Q is a manifold-with-corners, λ is a unimodular labelling, and c is a cohomology class.
result Classifies locally standard smooth actions of T up to equivariant diffeomorphisms.
We show the existence and orthogonality of wave operators naturally associated to a compatible Laplacian on a complete manifold with a corner of codimension 2. In fact, we prove asymptotic completeness i.e. that the image of these wave operators is equal to the space of absolutely continuous states of the compatible La…
In this paper we prove the propagation of singularities for the wave equation on differential forms with natural (i.e. relative or absolute) boundary conditions on Lorentzian manifolds with corners, which in particular includes a formulation of Maxwell's equations. These results are analogous to those obtained by the a…
Corners can be identified by a drum's sound spectrum.
problem Determining the presence of corners in a drum's shape from its sound.
method Proving spectral invariance of corners in domains with Lipschitz, piecewise smooth boundaries.
result Corners are uniquely determined by a drum's spectrum among domains with fixed genus.
One can define what it means for a compact manifold with corners to be a "contractible manifold with contractible faces." Two combinatorially equivalent, contractible manifolds with contractible faces are diffeomorphic if and only if their 4-dimensional faces are diffeomorphic. It follows that two simple convex polytop…
New method for constructing gluing parameterizations in geometric analysis.
problem Constructing gluing parameterizations for anti-self-dual connections.
method Using C1 maps of smooth Banach manifolds with corners. result Provides a method applicable to other nonlinear PDE problems.
We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stron…
Proves spacetime positive mass theorem with corners.
problem Proving a positive mass theorem for spacetime with corners.
method Deformation theorem with corner conditions, asymptotically flat initial data.
result Exterior end satisfies E≥∣P∣ in every dimension n≥3.