arXiv research

Generalizes complex manifolds to manifolds with corners and generalized corners.

In conventional Differential Geometry one studies manifolds, locally modelled on ${\mathbb R}^n$, manifolds with boundary, locally modelled on $[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\bf Man}\subset{\bf Man^b}\sub…

Manifolds with boundary and with corners form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. 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 $f:X\to\mathbb{R}$ 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.

Unified framework for gravity on manifolds with corners derived.

New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.

Defines products for fibered corners manifolds, generalizing resolutions.

Extends 4D cornered skein theory to surfaces, proving gluing formulas.

Extends algebraic geometry to include spaces with corners.

Paper proves rigidity for Penrose inequality on 3-manifolds with corners.

Generalizes differentiation under integral sign to submanifolds with corners.

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.

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary. As a main application, a normal fo…

The paper studies geometric obstructions for Fredholm conditions on manifolds with corners.

The paper solves obstructions to the Fredholm perturbation property for manifolds with corners.

Positive mass theorem for non-smooth metrics on flat manifolds with corners.

Defines s-manifolds and s-manifolds with corners for symplectic applications.

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.

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.

New framework for logarithmically divergent integrals on manifolds with corners.

The study of Einstein metrics with corners and boundaries.

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.

Develops a facewise formulation of AKSZ construction on manifolds with ordinary corners.

Introduces generalized products for pseudodifferential operators on manifolds with corners.

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.

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.

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

Generalizes moment-angle manifolds to arbitrary nice manifolds with corners.

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.

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.

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.

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.