Research examines metrics with positive scalar curvature in domains with corners.
problem Positive scalar curvature in domains with corners.
method Study of metrics with positive scalar curvatures in domains with corners.
result Possible extensions of positive scalar curvature to singular spaces.
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.
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.
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.
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.
We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows that a disk with two corners admits a conformal metric with constant Gauss curv…
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
Proves positive mass theorem for specific initial data sets with corners.
problem Initial data sets with corners and non-smooth boundaries.
method Axially symmetric, maximal, complete initial data sets with two ends, proving for axially symmetric, simply connected, maximal, complete initial data sets with two ends.
result Proves positive mass theorem for specific initial data sets with angular momentum and charges.
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.
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.
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…
New Calabi-Yau metrics constructed with detailed geometry at infinity.
problem Constructing complete Calabi-Yau metrics with specific properties.
method Weighted blow-up and Hölder spaces for Laplacian analysis.
result Examples of Calabi-Yau metrics with conical singularities and non-uniqueness of tangent cones.
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. Study on positive scalar curvature metrics with fundamental group constraints.
problem Nontriviality of space of metrics with positive scalar curvature on manifolds with torsionfree fundamental groups.
method Introduce stable metrics and round corners of manifolds, prove existence theorem.
result Existence of compact Spin 6-manifolds with infinite-dimensional rational homotopy groups for positive scalar curvature metrics.
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.
The paper constructs Ricci-positive metrics on complex manifolds.
problem Creating Ricci-positive metrics on connected sums of products of spheres.
method Two new technical theorems are introduced: a gluing construction for Ricci-positive manifolds with corners and a boundary deformation theorem.
result Ricci-positive metrics are constructed on the connected sum of products of arbitrarily many spheres.
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…
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 …
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.
The study defines differential forms and currents on orbifolds with corners.
problem Defining differential forms and currents on orbifolds with corners.
method Using the formalism of étale proper groupoids with corners, the authors provide constructions and proofs without orbifold charts.
result The Fréchet space of differential forms and the dual space of currents are independent of the chosen groupoid representation.
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.
Solves relative isoperimetric problem on polygonal domains, focusing on corners.
problem Relative isoperimetric problem on polygonal domains in R2. method Developed techniques for polygonal domains, with special attention to corners.
result Solved the relative isoperimetric problem for a square with a square corner removed.
Paper proves corner connection tiles can represent knots with fewer tiles.
problem Finding the minimum number of tiles for knot representation.
method Developed corner connection tiles and proved their efficiency.
result Corner connection tiles can represent knots with fewer tiles than traditional tiles.
We study the positive mass theorem for certain non-smooth metrics following P. Miao's work. Our approach is to smooth the metric using the Ricci flow. As well as improving some previous results on the behaviour of the ADM mass under the Ricci flow, we extend the analysis of the zero mass case to higher dimensions.
Smooth metrics satisfying Penrose inequality are necessarily smooth.
problem Rigidity of Penrose inequality with singular metrics.
method Showed suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality are smooth in specified coordinates.
result Smooth metrics satisfying Penrose inequality are necessarily smooth.
One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the Φ-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Mel…
Fourth-order problem on half-ball with corner behavior.
problem Fourth-order problem with corner behavior on half-ball.
method Conformal mapping to isolate corner effect.
result Gauss-Bonnet formula simplifies to constant term at corner.
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.
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.
Uniform estimates for elliptic problems near polygonal domains.
problem Proving uniform solvability estimates for elliptic problems near polygonal domains.
method Suitable conformal modification of the metric to make the union of domains a manifold with boundary and relative bounded geometry.
result Rounding off the corners of the limit polygonal domain.
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.
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 prove a Gauss-Bonnet theorem for (finite coverings of) moduli spaces of Riemann surfaces endowed with the McMullen metric. The proof uses properties of an exhaustion of moduli spaces by compact submanifolds with corners and the Gauss-Bonnet formula of Allendoerfer and Weil for Riemannian polyhedra.
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.
Currents with corners help count triangulations on surfaces.
problem Counting triangulations on surfaces with weighted vertices.
method Introduced currents with corners, studied their properties, and applied them to triangulation counting.
result The number of triangulations grows polynomially of degree 6g-6.
New tile types for knots and links reduce complexity.
problem Determining the minimum number of tiles needed for knot representations.
method Introduced new tile types and analyzed their impact on knot complexity.
result Corner tile number lies between tile number and 3 times tile number.
New heat trace coefficients reveal curvature effects in polygonal domains.
problem Understanding heat trace behavior in polygonal domains with curved corners.
method Local heat trace expansion through order t1/2, analyzing both Dirichlet and Neumann boundary conditions. result Sharp sign law for the Dirichlet angular factor of the first corner-curvature heat invariant.
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.
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…
Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
problem Gromov's flat corner domination conjecture and Stoker's conjecture for convex polyhedra.
method Same techniques applied to prove conjectures.
result Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
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…
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…
The study classifies graphs on surfaces with positive curvature properties.
problem Classifying graphs on surfaces with specific curvature properties.
method Using medial graphs and classification techniques.
result Complete classification of graphs on surfaces with positive Forman curvature and corner curvature.