Proves equivalence of two types of boundaries in metric spaces.
problem Proving equivalence of two types of boundaries in metric spaces.
method Analyzes and compares contracting and κ-Morse boundaries. result Proves equivalence of 1-Morse boundary and contracting boundary as topological spaces.
Under two boundary conditions, the generalized Atiyah-Patodi-Singer boundary condition and the modified generalized -Atiyah-Patodi-Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary.
Proves well-posedness for Einstein equations with specific boundary conditions.
problem Well-posedness of vacuum Einstein equations with twisted Dirichlet boundary conditions.
method Proves local-in-time well-posedness for the IBVP of the Einstein equations with specified conformal class and scalar densities.
result Proves well-posedness for the Einstein equations with twisted Dirichlet boundary conditions.
We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries via projective duality and observe the duality on boundary singularities. Moreove…
Proof of local well-posedness for a specific boundary condition in general relativity.
problem Initial boundary value problem in general relativity with umbilic boundary condition.
method Wave coordinates and key observation of momentum constraint validity for umbilic boundaries.
result Local well-posedness established for the initial boundary value problem.
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. Study on well-posedness of vacuum Einstein equations with specific boundary conditions.
problem Well-posedness of the initial boundary value problem for vacuum Einstein equations with geometric boundary conditions.
method Analysis of conformal-mean curvature boundary data, proving dense solution space and Holmgren-type uniqueness theorem.
result Linearized problem has a solution space with dense range in C∞, valid for general smooth linearized solutions. Maps with boundary definite fold points restrict manifold structure.
problem Restricting the global structure of manifolds with boundary.
method Introducing boundary special generic maps and deriving differential-topological restrictions.
result New results on non-singular extensions of special generic maps.
Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.
problem Understanding how Steklov eigenvalues respond to boundary changes.
method Analyzing smooth boundary perturbations of Steklov eigenvalues.
result Steklov eigenvalues are generically simple under such perturbations.
Characterizes relative hyperbolicity using Morse and contracting boundaries.
problem Understanding relative hyperbolicity in groups.
method Boundary-theoretic characterization using Morse and contracting boundaries.
result Non-elementary relatively hyperbolic groups have non-empty and compact Morse or contracting boundaries.
Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.
problem Optimal weighted Minkowski inequality for umbilical hypersurfaces with free boundary.
method Generalization of Xia's result to free boundary case in space forms.
result Free boundary version of optimal weighted Minkowski inequality in space forms.
Study critical metrics on manifolds with boundary using integral and boundary estimates.
problem Investigate geometry of critical metrics on compact manifolds with boundary.
method Use generalized Reilly's formula to derive integral and boundary estimates.
result Establish new boundary estimates for critical metrics of the volume functional.
Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.
problem Proving multiplicity one for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
method Developed existence and regularity theory for free boundary hypersurfaces with prescribed mean curvature, including Morse index bounds.
result Proved multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
Proves well-posedness for Einstein equations with specific boundary data.
problem Proving well-posedness for Einstein equations with Dirichlet boundary data.
method Local-in-time well-posedness proof for vacuum Einstein equations with specific boundary conditions.
result Proves well-posedness for Einstein equations with Dirichlet boundary data under convexity-type assumptions.
Study on static manifolds with boundary and rigidity of curvature.
problem Understanding the rigidity of scalar curvature and mean curvature on manifolds with boundary.
method Analyzing maps of scalar curvature in the interior and mean curvature on the boundary, discussing geometric properties of static manifolds.
result Classification and rigidity theorems for simple non-generic domains in space forms and Schwarzschild manifold.
This work connects the Hessian to the decision boundary complexity in neural networks.
problem Understanding the decision boundary complexity in high-dimensional input space.
method Characterizing the decision boundary using the Hessian top eigenvectors and analyzing the number of outliers.
result The number of outliers in the Hessian spectrum is proportional to the complexity of the decision boundary.
Local conditions on boundaries of C∞ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the hypersurface to be real analytic. This allows us to classify all real analytic generic bou…
We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instan…
Projective structures are mostly rigid at the boundary but some are not.
problem Boundary rigidity of projective structures.
method Investigation of projective structures on manifolds with boundary.
result Existence of non-rigid projective structures and characterization of them.
This paper classifies discrete conformal structures on surfaces with boundary.
problem Classifying discrete conformal structures on surfaces with boundary.
method Axiomatic approach ensuring good geometric structure, classification based on triangulation and axioms.
result Unified and generalized existing discrete conformal structures on surfaces with boundary.
Study boundary actions on CAT(0) spaces, proving topological freeness.
problem Understanding boundary actions on CAT(0) spaces.
method Verification of freeness of Myrberg points on boundaries.
result Large class of boundary actions are topologically free.
New boundary conditions improve Hamiltonian analysis in GR.
problem Improving Hamiltonian analysis in GR with IBVP.
method Presented and analyzed new boundary conditions.
result New boundary conditions lead to better Hamiltonian analysis.
This research extends topological recursion to hyperbolic surfaces with tight boundaries and conical defects.
problem Calculating volumes of hyperbolic surfaces with special boundaries.
method Generalized topological recursion to handle tight boundaries and conical defects.
result Weil-Petersson volumes are polynomial in boundary lengths for hyperbolic surfaces with tight boundaries and conical defects.
Study constant mean curvature surfaces with integrable boundary conditions.
problem Understanding surfaces with constant mean curvature under specific boundary conditions.
method Used generalized Weierstrass representation to determine potentials.
result Determined potentials for surfaces satisfying integrable boundary conditions.
New boundary for geodesic spaces captures Poisson boundary of mapping class groups.
problem Capturing the Poisson boundary of mapping class groups.
method Constructing a quasi-isometric invariant boundary for proper geodesic spaces.
result The Poisson boundary of mapping class groups can be realized on the κ-Morse boundary.
The paper studies rigid sphere packings on 3D manifolds with boundary.
problem Investigating rigid sphere packings on 3D manifolds with boundary.
method Introducing generalized sphere packings, proving rigidity, introducing combinatorial curvature flows.
result Generalized sphere packing metrics are determined by combinatorial scalar curvature.
The paper studies rigidity of sphere packings on 3D manifolds with boundary.
problem Rigidity of sphere packings on 3D manifolds with boundary.
method Introduced generalized Thurston's sphere packings and proved their rigidity properties.
result Generalized Thurston's sphere packings are locally determined by combinatorial scalar curvatures and cannot be deformed while keeping combinatorial Ricci curvatures fixed.
Random surfaces with boundary have predictable properties.
problem Understanding the statistical properties of random surfaces.
method Generating surfaces by gluing polygons and analyzing their genus and boundary components.
result Genus and boundary components of random surfaces follow a bivariate normal distribution.
Paper solves a mixed boundary value problem in space forms with umbilical boundaries.
problem Solving a partially overdetermined mixed boundary value problem in space forms.
method Generalizing previous results to domains with partial umbilical boundaries.
result A partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface.
BDSG generates samples on distribution boundaries, improving anomaly detection.
problem Difficulty in capturing multimodal supports and approximating distribution tails.
method Invertible Residual Network (IResNet) and Residual Flow (ResFlow) for density estimation; compound loss function for boundary samples.
result Competitive performance on synthetic and multimodal data compared to existing methods.
In this contribution we derive an explicit formula for the boundary non-crossing probabilities for Slepian processes associated with the piecewise linear boundary function. This formula is used to develop an approximation formula to the boundary non-crossing probabilities for general continuous boundaries. The formulas…
New method generalizes eigenvalue inequality to surfaces with boundaries.
problem Eigenvalue inequality for surfaces with boundaries.
method Generalized Rohleder's approach to differential forms, presenting Hodge-Laplacian spectrum.
result Obtained inequality for eigenvalues of Hodge-Laplacian and Dirichlet problems.
Proves new theorems for manifolds with boundary.
problem No boundary conditions for manifolds.
method Generalizes Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
result Proof of new theorems for even and odd dimensional manifolds with boundary.
This paper is devoted to a proof of a generalized Ray-Singer conjecture for a manifold with boundary (the Dirichlet and the Neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary). The Ray-Singer conjectur…
The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
method Introducing combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings.
result Proves longtime existence and global convergence of combinatorial curvature flows.
Paper presents an algorithm to analyze DGNNs by identifying generative boundaries and sampling.
problem Challenges in identifying the generation mechanism of DGNNs.
method Explorative sampling algorithm to analyze DGNNs by defining generative boundaries and optimizing essential set.
result Empirically reveals the characteristics of DGNNs' internal layers and finds more homogeneous samples.
The paper classifies CMC free boundary hypersurfaces in rotational domains.
problem Existence and uniqueness of free boundary constant mean curvature hypersurfaces in rotational domains.
method Classification and construction of CMC free boundary hypersurfaces under specific conditions.
result Classification of CMC free boundary hypersurfaces as topological disks or annuli.
Proves another theorem for odd dimensional manifolds with boundary.
problem Spectral Einstein functional on odd dimensional manifolds with boundary.
method Proof of a theorem using the Dirac operator.
result Another general Dabrowski-Sitarz-Zalecki type theorem proved.
An n-dimensional manifold M (n≥3) is called {\it generalized graph manifold} if it is glued of blocks that are trivial bundles of (n−2)-tori over compact surfaces (of negative Euler characteristic) with boundary. In this paper two obstructions for generalized graph manifold to be nonpositively curved are des…
We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contains local elliptic boundary conditions in the sense of Lopatinskij and Shapiro as well as the Atiyah-Patodi-Singer bo…
In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with C2 boundary. This appears to be the first boundary Schwarz lemma for general holomorphic self maps that requires no strong pseudoconvexity or finite type assumptions. T…
DeepDIG generates samples near decision boundaries of deep neural networks for better understanding.
problem Limited knowledge of how deep neural networks make decisions.
method Adversarial example generation to create samples near decision boundaries.
result Characterized decision boundaries of various deep neural network models.
Boundary Dehn twists become trivial after abelianization.
problem Understanding the behavior of boundary Dehn twists in smooth mapping classes.
method Constructing diffeomorphisms and showing commutators represent boundary Dehn twists.
result Boundary Dehn twists become trivial after abelianization.
Proves two theorems on odd-dimensional manifolds with boundary.
problem Proving theorems on manifolds with boundaries.
method Proof of theorems using mathematical techniques.
result Proved the general Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki type theorems.
A guide for solving first-order elliptic boundary value problems.
problem Solving first-order elliptic boundary value problems on manifolds.
method Operator methods and general elliptic boundary conditions.
result Characterization of a new subclass of elliptic boundary conditions.
We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local. We show the equivalence of various characterisations of elliptic boundary conditions and demonst…
For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,∂M) where 3≤(n+1)≤7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates t…
In [3], Connes found a conformal invariant using Wodzicki's 1-density and computed it in the case of 4-dimensional manifold without boundary. In [14], Ugalde generalized the Connes' result to n-dimensional manifold without boundary. In this paper, we generalize the results of [3] and [14] to the case of manifolds wit…