Study on the geometry of limit spaces of manifolds with boundary.
problem Understanding the geometry of limit spaces of manifolds with boundary.
method Developed infinitesimal geometry for limit spaces under curvature and diameter bounds.
result Determined the infinitesimal structure and Hausdorff dimensions of boundary singular sets.
For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geomet…
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
The paper explores the geometry and topology of DNN decision boundaries.
problem Understanding the geometric and topological properties of DNN decision boundaries.
method Differential geometry and the Gauss-Bonnet-Chern theorem.
result Computed the Euler characteristics of compact decision boundaries.
In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compac…
We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with c…
Generalizes Bestvina's Z-boundaries to coarse Z-boundaries.
problem Establishing properties of Z-boundaries for groups. method Introducing a new concept of a 'coarse Z-boundary' and proving theorems about it. result Admitting a coarse Z-boundary is a pure quasi-isometry invariant. The paper establishes lower bounds on injectivity radius and constructs metrics with bounded geometry.
problem Establishing lower bounds on the normal injectivity radius of hypersurfaces and constructing metrics with bounded geometry on manifolds with boundary.
method Pointwise lower estimates and constructions of metrics with bounded geometry.
result The construction of metrics with bounded geometry on arbitrary manifolds with boundary.
Study on static perfect fluid space-time geometry and boundary estimates.
problem Investigate the geometry and boundary properties of static perfect fluid space-time.
method Used generalized Reilly's formula to establish geometric inequalities and boundary estimates.
result Obtained new boundary estimates involving the Brown-York mass and first eigenvalue of the Jacobi operator.
Symplectic structure found on projective structures on surfaces with boundary.
problem Deformation space of projective structures on surfaces with boundary.
method Natural symplectic structure on the space, integrating the Adler-Gelfand-Dikii-space of the boundary.
result Space is a Hamiltonian space for the symplectic groupoid.
Study connects boundary geometry to symbol of Dirichlet-to-Neumann operator.
problem Determining geometric data from boundary symbol of connection Laplacian.
method Analyze symbol of Dirichlet-to-Neumann operator associated with connection Laplacian.
result Geometric data on boundary and normal derivatives are determined by symbol.
This work uses tropical geometry to understand neural network decision boundaries.
problem Characterizing neural network decision boundaries with piecewise linear activations.
method Tropical geometry applied to a simple neural network model.
result Decision boundaries are a subset of a tropical hypersurface related to a polytope formed by zonotopes.
Study on quasi-Einstein manifolds with boundary estimates and inequalities.
problem Understanding the geometry of compact quasi-Einstein manifolds with boundary.
method Sharp boundary estimates and characterization theorems for quasi-Einstein manifolds.
result New geometric inequalities and boundary estimates for quasi-Einstein manifolds.
We consider a class of complete Kahler manifolds with a strictly pseudoconvex boundary at infinity. After studying its asymptotic geometry, we formulate a conjecture in the Kahler-Einstein case relating the bottom of spectrum to the CR geometry on the boundary. We prove some partial results.
Study compares manifolds with boundary under weighted Ricci curvature bounds.
problem Understand geometric properties of manifolds with boundary under lower weighted Ricci curvature bounds.
method Use lower N-weighted Ricci curvature bounds with ε-range to study comparison geometry. result Conclude splitting theorems and comparison geometric results for inscribed radius, volume, and eigenvalues.
Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator Δ:Hk+1(M)∩H01(M)→Hk−1(M), k∈N0, invertible?" We consider also the case of mixed boundary conditions. The study of this main question lea…
After introducing the different boundary geometries of rank one symmetric spaces, we state and prove Fried's theorem in the general setting of all those geometries: a closed manifold with a similarity structure is either complete or the developing map is a covering onto the Heisenberg-type space deprived of a point.
This article is based on a lecture at the Journal of Differential Geometry Conference, Harvard 2017. We discuss closed and torsion-free G2-structures on a 7-manifold with boundary, with prescribed 3-form on the boundary. Much of the article is based on an observation that there is an intrinsic notion of "mean co…
Study the geometry of gas giant planets to infer their internal structure.
problem Determine the interior structure of gas giant planets using boundary data.
method Geometric analysis of Riemannian manifolds with conformal blow-up at the boundary.
result The interior structure of a gas giant is uniquely determined by different types of boundary data.
Three functors link Lorentzian geometry concepts.
problem Finiteness results, singularity theorems, boundary constructions.
method Review of three functors from Lorentzian categories.
result Novel functor from ordered measure spaces to Lorentzian pre-length spaces.
Given two compact Riemannian manifolds with boundary M1 and M2 such that their respective boundaries Σ1 and Σ2 admit neighborhoods Ω1 and Ω2 which are isometric, we prove the existence of a constant C, which depends only on the geometry of Ω1≅Ω2, such that ∣σk(M1)−σk(M2)∣≤C for eac…
In this paper, we consider the problem of building a conformal boundary, embedding a pseudo-Riamnnian manifold as an open subset of a bigger one. We get first results about conformal maximality. We also show that in dimension ≥3, there are rigidity properties for the topological boundary of such a conformal embed…
The paper studies static manifolds with boundary and their properties.
problem Properties of static manifolds with boundary.
method Theorems relating topology and geometry, isoperimetric inequality, uniqueness theorems.
result Characterization of the round ball in Euclidean 3-space as the only scalar-flat static manifold with mean-convex boundary.
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem (P,C) is equivalent to the uniform regularity of the natural family (Px,Cx) of associated boundary value …
Researchers solve 3D cube complex boundary rigidity problem.
problem Determining the combinatorial type of a 3D CAT(0) cube complex from boundary distances.
method Discrete version of boundary rigidity problem, focusing on CAT(0) cube complexes.
result The combinatorial type of a finite CAT(0) cube complex can be reconstructed from its boundary distances.
Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.
problem Finding a conformal metric with zero scalar curvature and prescribed boundary mean curvature.
method Construction of local test functions to resolve open cases and establish new solvability conditions.
result Established new solvability conditions for the problem.
New surfaces defy traditional pants decompositions.
problem Creating surfaces without bounded pants decompositions.
method Constructing quasiconformally homogeneous hyperbolic Riemann surfaces.
result Found surfaces with infinite type and compact boundary components that defy pants decompositions.
Potential theory extended to Gromov hyperbolic spaces.
problem Extending potential theory to a new class of spaces.
method Unified framework for Gromov hyperbolic metric measure spaces.
result Boundary Harnack inequalities and complete classification of positive harmonic functions.
The article explores the fundamental gap in Bakry-Emery geometry.
problem The fundamental gap in Bakry-Emery geometry.
method Recalled Bakry-Emery geometry and connected eigenvalues with boundary conditions. Showed a connection between fundamental gap and Bakry-Emery geometry.
result Presented key ideas in Andrews's and Clutterbuck's proof of the fundamental gap conjecture.
Study contact geometry of symplectic divisors, invariant under specific transformations.
problem Understanding contact structures on symplectic divisors and their boundaries.
method Invariant analysis of contact structures under toric and interior blow-ups/blow-downs, open book decomposition construction.
result Contact structure on divisor boundaries is invariant under specified transformations.
Study of free boundary minimal Möbius bands in spherical caps.
problem Characterizing minimal surfaces with free boundary in spherical caps.
method Analyzing spectral properties and geometric constraints.
result Proves that any free boundary minimal Möbius band in spherical caps must be intrinsically rotationally symmetric.
Recovering manifold geometry from geodesic intersections.
problem Recovering the geometry of a Riemannian manifold from geodesic intersection lengths.
method Applying stitching data to solve the delayed collision data problem.
result Geometry of the manifold can be recovered from geodesic intersection lengths.
This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded geometry and we recover their singular set as the Gromov boundary but also as the Marti…
Researchers find multiple ways to deform manifolds with specific curvature properties.
problem Finding distinct conformal deformations of manifolds with boundary conditions.
method Using bifurcation results from Case, Moreira, and Wang, the researchers construct geometrically distinct solutions.
result There are multiple solutions to the conformal deformation problem in a finite set of dimensions.
New examples show positive scalar curvature metrics on manifolds with boundary that cannot be extended.
problem Positive scalar curvature metrics on manifolds with boundary that cannot be extended.
method Analytic techniques related to the prescribed scalar curvature problem in conformal geometry.
result Obstruction to positivity of conformal Laplacians given by a real-valued ξ-invariant.
We prove a Feynman-Kac formula for differential forms satisfying absolute boundary conditions on Riemannian manifolds with boundary and of bounded geometry. We use this to construct L2 harmonic forms out of bounded ones on the universal cover of a compact Riemannian manifold whose geometry displays a positivity prop…
Let M be a smooth manifold with boundary ∂M and interior M. Consider an affine connection ∇ on M for which the boundary is at infinity. Then ∇ is projectively compact of order α if the projective structure defined by ∇ smoothly extends to all of M in a spec…
In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectra…
On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature. Then we discuss rel…
Symplectic structures on Teichmüller spaces for surfaces with ideal boundary.
problem Understanding symplectic structures on Teichmüller spaces for surfaces with boundary.
method Defined natural symplectic structures and proved Hamiltonian properties.
result Explicit formula for Hill potential and global Darboux coordinates.
Extends static vacuum metrics with specific boundary conditions.
problem Proving the existence of static vacuum metrics with prescribed boundary data.
method Introducing static regular types (I) and (II), showing local well-posedness, and confirming Bartnik's conjecture.
result Confirms Bartnik's static vacuum extension conjecture for a broad range of boundary conditions.
Let (M,g) be a compact Einstein manifold with smooth boundary. We consider the spectrum of the p form valued Laplacian with respect to a suitable boundary condition. We show that certain geometric properties of the boundary may be spectrally characterized in terms of this data where we fix the Einstein constant.
Unique hyperbolic manifolds identified by boundary pleating.
problem Identifying hyperbolic manifolds from their boundary pleating.
method Used pleating measured lamination on the boundary of convex cores.
result Convex co-compact hyperbolic manifolds are uniquely determined by their pleating lamination.
Deep NURBS improves PINNs for solving PDEs on arbitrary geometries.
problem Solving partial differential equations on complex geometries with physics constraints.
method Combines admissible NURBS parametrizations and PINN solver for arbitrary geometries.
result High convergence rate and accuracy for most PDEs using Deep NURBS.
Two different spacetimes can mimic each other's boundary measurements.
problem Lorentzian Calderón problem
method Counterexample construction
result Non-isometric spacetimes can have identical boundary measurements
Sharp stability estimate for tensor tomography in non-positive curvature.
problem Stability estimate for tensor tomography on manifolds with non-positive curvature.
method Pestov identity with localized frequency boundary term.
result Stability estimate of the form L2↦HT1/2. Study on hyperbolic manifolds with special boundaries.
problem Characterizing geometric properties of hyperbolic manifolds with specific boundaries.
method Analyzing asymptotically hyperbolic manifolds with a horospherical boundary.
result Derived geometric formulas for the model case of horoballs and their complements.