The paper proves local and long-term existence of Ricci de Turck flow on incomplete edge manifolds.
problem Proving existence of Ricci de Turck flow on incomplete edge manifolds.
method Careful analysis of the Lichnerowicz Laplacian and the Ricci de Turck flow equation.
result Local and long-term existence of Ricci de Turck flow on incomplete edge manifolds.
Study on eta and rho invariants on incomplete edge spaces.
problem Existence and properties of eta and rho invariants on incomplete edge spaces.
method Microlocal analysis of heat kernel asymptotics and Atiyah-Patodi-Singer arguments.
result Derivation of an Atiyah-Patodi-Singer index theorem for incomplete edge spaces.
We derive a formula for the index of a Dirac operator on a compact, even-dimensional incomplete edge space satisfying a "geometric Witt condition". We accomplish this by cutting off to a smooth manifold with boundary, applying the Atiyah-Patodi-Singer index theorem, and taking a limit. We deduce corollaries related to …
Study Dirac operators on incomplete cusp edge spaces, proving self-adjointness and Fredholm properties.
problem Analyzing Dirac operators on complex geometric spaces.
method Construct heat kernel, prove self-adjointness and Fredholm properties, establish index formula.
result Proved Dirac operators are essentially self-adjoint and Fredholm.
The article proves long-time existence and convergence of the edge Yamabe flow.
problem Analyzing the normalized Yamabe flow on incomplete edge singularities.
method Novel maximum principle results and uniform bounds established without barrier functions or Krylov-Safonov estimates.
result Long-time existence and convergence of the edge Yamabe flow.
Defines odd Pfaffian form for odd-dimensional manifolds, proving Chern-Gauss-Bonnet formula.
problem Proving Chern-Gauss-Bonnet formula for incomplete edge singularities.
method Defining odd Pfaffian form through curvature tensor, proving formula for various metrics.
result Proves intrinsic Chern-Gauss-Bonnet formula for edge singularities and complete manifolds.
Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schauder-type estimates for the heat operator on certain Hölder spaces …
We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. Thi…
We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extend…
Let (X,g) be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of X, as well as the associated spac…
We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided …
GRCN improves GCNs by predicting missing edges and revising weights.
problem Sub-optimal solutions due to incomplete and noisy real-world graphs.
method Introduces a GCN-based graph revision module for predicting missing edges and revising weights.
result GRCN consistently outperforms strong baseline methods, especially on incomplete graphs.
We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescrib…
Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove…
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.
Study shows long-term flow on special manifolds with positive Yamabe constant.
problem Analyzing long-time behavior of Yamabe flow on singular spaces.
method Formulated axioms for long-time existence, used parabolic Moser iteration for bounds.
result Established long-time existence of normalized Yamabe flow on specified manifolds.
Framework predicts logical queries on incomplete knowledge graphs.
problem Complex logical queries involving multiple unobserved edges and entities.
method Low-dimensional embeddings and learned geometric operations.
result Efficient predictions with linear time complexity in query variables.
New flow preserves singularities on incomplete manifolds.
problem Evolve incomplete manifolds with bounded curvature.
method Construct Ricci de Turck flow uniformly equivalent to initial metric.
result Any incomplete manifold can be evolved for a short time.
Algorithm reconstructs spreading model parameters from incomplete data.
problem Reconstructing unknown transmission probabilities from partial observation data.
method Dynamic message-passing algorithm for incomplete spreading data.
result Efficient algorithm reconstructs parameters of spreading models.
The paper proves pseudolocality theorems for Ricci flows on incomplete manifolds.
problem Pseudolocality of Ricci flows on incomplete manifolds.
method Proves pseudolocality theorems for Ricci flows under specific curvature and isoperimetric conditions.
result Constructs solutions of Ricci flow in balls with pseudolocality property.
Improved prediction accuracy in matrix factorization using graph-based priors.
problem Graph side-information may not align with latent-feature relations in matrix completion.
method Identify and remove 'contested' edges using graphical lasso approximation, maintaining linear scalability.
result Improved prediction accuracy with fewer graph edges, demonstrating the often inaccurate nature of graph side-information.
Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.
problem Control of Lorentzian lengths of limit curves in incomplete metric spaces.
method Proves limit curve theorem for incomplete metric spaces and applies to null distance.
result Strong control on Lorentzian lengths of limit curves in Sormani and Vegas' null distance.
Bayesian method infers transition matrices from incomplete graph data with topological constraints.
problem Inference of transition matrices from incomplete graph data with topological constraints.
method Bayesian approach using repeated interactions and a topological prior.
result Higher accuracy in inferring transition probabilities, improving downstream tasks.
Extends cohomology to incomplete Riemannian manifolds.
problem Cohomology of harmonic forms on incomplete Riemannian manifolds.
method Abstract setting of Hilbert complexes.
result Geometric applications to Thom-Mather spaces.
Proposes learning graph structure and GCN parameters together.
problem Inability of GNNs to use graphs that are incomplete or corrupted.
method Solves a bilevel program to learn a discrete graph structure.
result Outperforms related methods in experiments.
Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete …
The paper proves positivity preservation and self-adjointness for Schrödinger operators on incomplete Riemannian manifolds.
problem Positivity preservation and self-adjointness for Schrödinger-type operators on incomplete Riemannian manifolds.
method Control of potential behavior near the Cauchy boundary, essential self-adjointness proof, core of smooth compactly supported functions.
result Positivity preservation and essential self-adjointness of Schrödinger operators on Lp functions on incomplete Riemannian manifolds. Formula derived for G2-manifolds, showing moduli spaces are incomplete.
problem Incompleteness of moduli spaces for G2-manifolds. method Derived a formula for the energy of paths in moduli spaces, provided conditions for finite energy and length.
result Compact G2-manifolds produced by the generalised Kummer construction have incomplete moduli spaces. Sharp bounds on heat kernel derivatives on incomplete manifolds.
problem Extending bounds on heat kernel derivatives to incomplete Riemannian manifolds.
method Analyzing heat kernels on incomplete Riemannian manifolds with conservative and non-conservative vector fields.
result Sharp bounds on all orders of heat kernel derivatives are established for incomplete manifolds.
We clarify the relationship between the null geodesic completeness of an Einstein Lorentz manifold and its conformal Kobayashi pseudodistance. We show that an Einstein manifold has at least one incomplete null geodesic if its pseudodistancfe is nontrivial. If its pseudodistance is nondegenerate, all of its null geodesi…
The paper proves density and positive mass theorems for incomplete manifolds.
problem Proving density and positive mass theorems for manifolds with incomplete ends.
method Using harmonic asymptotics and quantitative positive mass theorem improvements.
result Improved quantitative positive mass theorem in dimensions 3 to 7.
Paper explores exact recovery of communities in weighted graphs using Gaussian and exponential distributions.
problem Exact recovery of communities in weighted graphs with Gaussian and exponential distributions.
method Introduces a new semi-metric to describe conditions for exact recovery and analyzes conditions for both complete and incomplete graphs.
result Necessary and sufficient conditions for exact recovery are asymptotically tight and applicable to both complete and incomplete graphs.
The paper characterizes stochastic incompleteness in Riemannian manifolds.
problem Stochastic incompleteness of Riemannian manifolds and its characterization.
method Characterization through solutions to nonlinear parabolic equations.
result Stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to certain nonlinear parabolic equations.
A fast framework for image deconvolution with incomplete observations.
problem Deconvolution with unknown boundaries is slow and artifacts-prone.
method Diagonalization of convolution operators, iterative pixel estimation and deconvolution.
result Framework allows fast deconvolution with unknown boundaries.
We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is clo…
The paper studies cohomology on incomplete manifolds and stratified spaces.
problem Analyzing cohomology groups on incomplete Riemannian manifolds and stratified spaces.
method Proves injective/surjective maps between Lp and L2 cohomology groups under certain conditions. result Injective/surjective maps between Lp and L2 cohomology groups are established. In this article we extend the Gallot-Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over such a manifold admits a parallel symmetric 2-tensor then it is incomplete and has non zero constant curvature. An application of this result to the existence of metrics with distinct Le…
This work completes Chern-Ricci flow on complex manifolds with incomplete data.
problem Existence and behavior of Chern-Ricci flows on complex manifolds.
method Analyzes the flow and potential flow on complex manifolds with incomplete initial data.
result Obtains existence results for Chern-Ricci flows and Kähler-Einstein metrics.
Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle 2π(1−α) for α∈(0,1). In this paper we study how the existence of such Kähler-Einstein metrics depends on α. We show that in the negative s…
Study geometric quantum confinement on special incomplete Riemannian manifolds.
problem Characterize quantum confinement on Grushin-type manifolds.
method Constant-fibre direct integral scheme combined with Weyl's analysis.
result Fully characterizes essential self-adjointness of Laplace-Beltrami operator.
Gradient descent near stability threshold exhibits sharpness oscillations.
problem Understanding sharpness behavior near stability threshold in non-Euclidean norms.
method Interpreted EoS through Directional Smoothness and generalized sharpness under arbitrary norms.
result Non-Euclidean GD with generalized sharpness shows sharpness oscillations near 2/η. The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
Proves mass theorem for manifolds with arbitrary ends.
problem Proving the positive mass theorem for manifolds with various ends.
method Quantitative analysis of scalar curvature on manifolds with arbitrary ends.
result Proves the positive mass theorem for a wide class of manifolds.
Study special Lagrangian submanifolds with edge singularities using elliptic theory.
problem Characterize the moduli space of deformations of special Lagrangian submanifolds with edge singularities.
method Use elliptic theory for edge-degenerate differential operators on singular manifolds.
result Obtain a general theorem describing the local structure of the moduli space.
Researchers prove formulas for flag area measures, extending previous work.
problem Proving additive kinematic formulas for flag area measures.
method Introducing an algebraic framework to compute these formulas explicitly.
result Existence and explicit computation of additive kinematic formulas for flag area measures.
Withdrawn due to an incompleteness of the main results.
New approach solves Calabi problem on manifolds with edge-cone singularities.
problem Solving the Calabi problem on manifolds with edge-cone singularities.
method Proposes a new approach using a good reference metric and equivalent equations with different reference metrics.
result Extends methods from smooth settings to edge settings, generalizing to multiple hypersurfaces.
New compact ECS manifolds with rank 2 discovered, differing from previous rank 1 examples.
problem Finding new compact ECS manifolds with rank 2.
method Constructing new examples of compact pseudo-Riemannian manifolds with parallel Weyl tensor, rank 1 or 2.
result New compact ECS manifolds of rank 2, locally homogeneous, and geodesically incomplete.