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.
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.
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.
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…
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.
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.
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.
New method reveals corners of drum shapes.
problem Determining the shape of drum corners from its sound.
method Locality principle and calculations of heat kernels.
result Corners are spectral invariants of the Laplacian.
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.
TacticAI helps football coaches improve tactics by analyzing corner kicks.
problem Developing effective responses to rival teams' tactics algorithmically.
method TacticAI combines predictive and generative components to suggest player setups and position adjustments.
result TacticAI's model suggestions are favored over existing tactics 90% of the time by football domain experts.
Let Ω0 be a polygon in $\RR^2$, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that $Ω_\e$ is a family of surfaces with $\calC^\infty$ boundary which converges to Ω0 smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov …
In 1996, Kirk Lancaster and David Siegel investigated the existence and behavior of radial limits at a corner of the boundary of the domain of solutions of capillary and other prescribed mean curvature problems with contact angle boundary data. In Theorem 3, they provide an example of a capillary surface in a unit disk…
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.
Formula for Laplacian determinants on polygonal domains with slits.
problem Determining the ζ-regularized determinant of the Laplacian on polygonal domains with slits. method Patchwork method for heat trace asymptotics, comparison formula for smooth conformal metrics.
result Polyakov-Alvarez type formula for Laplacian determinants on polygonal domains with slits.
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…
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. 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.
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 …
Local corner-factor conjecture for Neumann jump determinants supported by models.
problem Determining the determinant of Neumann jump operator on piecewise curves.
method Formulated conjecture, supported by three model calculations, and discussed connections.
result Support for the conjectural determinant formula \(\Det_{\angle}'\cN = \frac{\length(\partial P)}2 \prod_{j=1}^Nα_j^{-1/2}\).
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.
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.
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.
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. 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.
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.
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.
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.
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…
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.
Real blow-up, including inhomogeneous versions, of boundary faces of a manifold (with corners) is an important tool for resolving singularities, degeneracies and competing notions of homogeneity. These constructions are shown to be particular cases of `generalized boundary blow-up' in which a new manifold and blow-down…
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…
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…
This paper makes a formal study of asymptotically hyperbolic Einstein metrics given, as conformal infinity, a conformal manifold with boundary. The space on which such an Einstein metric exists thus has a finite boundary in addition to the usual infinite boundary and a corner where the two meet. On the finite boundary …
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.
For every connected manifold with corners we use a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space),…
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…
Smooth analog of Gromov's dihedral rigidity for 3D weakly convex domains.
problem Rigidity of 3D weakly convex domains with nonnegative scalar curvature.
method Capillary minimal surfaces and foliations with nonnegative mean curvature.
result Smooth analog of Gromov's dihedral rigidity for 3D weakly convex domains.
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.
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.
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…
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.
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.
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…