The paper studies solutions to the Yamabe equation on asymptotically flat manifolds and their behavior at infinity.
problem Behavior of solutions to the Yamabe equation on asymptotically flat manifolds.
method Establishing asymptotic behavior near isolated singularities and using appropriate flatness conditions.
result Positive solutions on asymptotically flat manifolds of flatness order at least (n-2)/2 converge to fundamental solutions or radial Fowler solutions at infinity.
Flat solutions don't guarantee generalization for logistic loss in neural networks.
problem Proving flat solutions imply generalization for logistic loss in neural networks.
method Analyzing overparameterized two-layer ReLU networks with univariate input under logistic loss.
result Flat solutions enjoy near-optimal generalization bounds within uncertain sets but can still overfit at infinity.
Ancient solutions found for a specific flow on symplectic half-flat structures.
problem Existence of solutions for a particular geometric flow.
method Analyzes Type IIA flow on symplectic half-flat SU(3)-structures.
result Existence of ancient, immortal, and eternal solutions under suitable conditions.
New exact spherically symmetric vacuum solutions found in Finsler gravity.
problem Finding exact vacuum solutions in Finsler gravity.
method Spherically symmetric, asymptotically flat Berwald spacetimes solved for Finsler gravity vacuum equation.
result Only one class of spherically symmetric Berwald spacetimes is compatible with asymptotic flatness and a well-defined causal structure.
The paper constructs a new Ricci-flat metric on almost abelian Lie groups.
problem Finding Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups.
method Constructing left-invariant metrics on almost abelian Lie groups, focusing on dimensions four or higher.
result A new Ricci-flat metric that generalizes the Petrov solution to higher dimensions.
This is the second part of the investigation started in [Stationary solutions and asymptotic flatness I]. We prove here that Strongly Stationary ends having cubic volume growth are Weakly Asymptotically Flat. Combined with the results of the previous paper this shows that Strongly Stationary ends are Asymptotically Fla…
Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.
problem Constructing Abelian magnetic zero-modes on flat spacetime.
method Establishing a correspondence between vortex equations and harmonic spinors on the Nappi-Witten space.
result Explicit solutions of a twisted Dirac equation induce harmonic spinors on Minkowski space.
Noncompact Ricci-flat solutions have infinite unstable dimensions.
problem Understanding unstable dimensions of noncompact Ricci-flat solutions.
method Derived sufficient conditions for infinite-dimensional unstable manifolds.
result Noncompact Ricci-flat solutions have uncountably many unstable perturbations.
Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.
problem Behavior of flat flow solutions on planar flat torus.
method Sharp quantitative Alexandrov inequality derivation for periodic smooth sets.
result Flat flows converge to specific configurations exponentially fast.
The well-known Funk metric F(x,y) is projectively flat with constant flag curvature K=-1/4 and the Hilbert metric H(x,y):=(F(x,y)+F(x,-y))/2 is projectively flat with constant curvature K=-1. These metrics are the special solutions to Hilbert's Fourth Problem. In this paper, we construct a non-trivial R-flat spray usin…
Study classifies solutions to specific equations on half-space and ball.
problem Classifying nonnegative solutions to Q-flat and constant T-curvature equations. method Introduced a biharmonic Poisson kernel and derived its explicit representation formula.
result Established classification theorems for solutions on R+n+1 and Bn+1. Solves geodesic equations on specific metrics types.
problem Finding explicit solutions for geodesic equations on Eguchi-Hanson metrics.
method Analyzes geodesic equations under different constant conditions.
result Explicit solutions not yet available for geodesic equations when only ψ is constant. We show that supersymmetric flux vacua with intermediate SU(2) structure is closely related to some special classes of half-flat structures. More concretely, solutions of the SUSY equations IIA possess a symplectic half-flat structure, whereas solutions of the SUSY equations IIB admit a half-flat structure which is in …
Paper classifies solutions to oriented associativity equations on flat F-manifolds.
problem Classifying quasi-homogeneous formal power series solutions.
method Introducing monodromy local moduli and solving Riemann-Hilbert-Birkhoff problem.
result Formal germs of flat F-manifolds are convergent if not strictly doubly resonant.
Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on complex Lie groups. Both flat and non-flat cases are discussed in detail.
Study on solutions near isolated singularities in 6D Yamabe equation.
problem Behavior of solutions near isolated singularities in 6D Yamabe equation.
method Analyze asymptotic behavior of local solutions in non-conformally flat metrics.
result Solutions are asymptotically close to Fowler solutions in 6D.
Dubrovin duality connects two F-manifolds on the universal curve.
problem Connecting two F-manifolds on the universal curve.
method Proving natural extension of Dubrovin dual to F-manifolds with compatible flat connection.
result Equips the universal curve with two F-manifolds with compatible flat structure.
The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.
problem Understanding ancient solutions to curvature flows in bounded and unbounded regions.
method Constructing pancake-like and sausage-like ancient compact solutions.
result Ancient solutions to curvature flows in bounded and unbounded regions.
Zeroth-order methods favor flat minima in machine learning.
problem Finding solutions with small Hessian trace in optimization.
method Zeroth-order optimization with two-point estimator.
result Zeroth-order optimization converges to flat minima.
Study shows uniqueness of solutions on complex manifolds without requiring solution decay.
problem Uniqueness of solutions to Monge-Ampere equation on complex manifolds.
method Caccioppoli inequality techniques applied to Kähler manifolds with sub-quadratic volume growth.
result Uniqueness of bounded C1,1 solutions to Monge-Ampere equation without decay requirement. Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifo…
New method constructs flat initial data for Einstein's equations.
problem Constructing asymptotically flat initial data for Einstein's equations.
method Explicit solution operators with localization properties.
result Improved decay rate and nontrivial initial data construction.
Study cylindrical symmetric Finsler metrics that are projectively flat.
problem Characterize Finsler metrics that are projectively flat.
method Solve the system of differential equations for cylindrical symmetric Finsler metrics.
result Provide a family of solutions for the projectively flat Finsler metrics.
We prove that Maxwell fields of asymptotically flat solutions of the Einstein-Maxwell equations inherit the stationarity of the metric.
An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…
Local equivalence shown between specific distributions and flat Cartan distribution.
problem Establishing local equivalence between specific distributions and flat Cartan distribution.
method Change of coordinates mapping specific distributions to flat Cartan distribution.
result Local equivalence between maximally symmetric (2,3,5)-distributions and flat Cartan distribution. In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold …
Seiberg-Witten theory connects 3-manifold connections to spinor solutions.
problem Constructing Seiberg-Witten moduli spaces and understanding blow-up sets.
method Interpreting flat PSL(2;R)-connections as Seiberg-Witten solutions with two spinors.
result Explicit examples of Seiberg-Witten moduli spaces constructed.
Anomaly flow studied on flat and non-flat nilmanifolds.
problem Analyzing the Anomaly flow on nilmanifolds.
method Examined with respect to Hermitian connections, focusing on flat and non-flat cases.
result General solutions and qualitative behavior of the Anomaly flow on nilmanifolds.
Study totally real flat minimal surfaces in hyperquadric.
problem Characterize totally real minimal surfaces in complex hyperquadric.
method Analyze geometric properties and use harmonic sequences.
result Classify totally real flat minimal surfaces for N=4, 5, 6.
New results show flat minima in neural networks suffer from high dimensionality.
problem Flat minima in neural networks generalize poorly in high dimensions.
method Theoretical analysis of two-layer ReLU networks with multivariate inputs.
result Flat minima lead to exponentially slower convergence in high dimensions.
We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé's system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dep…
New model shows SGD can prefer sharp or flat solutions based on label noise.
problem Understanding SGD's preference for flat or sharp solutions during training.
method Solved an analytically solvable model to explore SGD behavior.
result Data distribution determines sharpness at convergence; isotropic label noise leads to flat minimum preference.
This paper has several goals. The first idea is to study the geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness in a modern and rigorous way. Although the idea is not new, our main Theorems about flatness introduce a different point of view in Differential Geometry. The…
We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in Cm with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in Cm with the pseudo-Euclidean metric is flat if the H…
New gravitational solitons and infinite topological manifolds found.
problem Finding new gravitational solitons and Riemannian manifolds.
method Space-periodic solutions of Einstein equations, quotients, Wick rotation.
result Complete Ricci flat Riemannian manifolds of infinite topological type.
Study shows how flat flow solutions in 2D converge to disks.
problem Understanding the asymptotics of area-preserving mean curvature flow in 2D.
method Analyzes flat flow solutions starting from bounded sets of finite perimeter.
result Flat flow solutions converge to a union of equally sized disks with exponential rate.
In this work, the dual flatness, which is connected with Statistics and Information geometry, of general (α,β)-metrics (a new class of Finsler metrics) is studied. A nice characterization for such metrics to be dually flat under some suitable conditions is provided and all the solutions are completely determined. By …
We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.
We classify (up to local isometry) the maximally supersymmetric solutions of the eleven- and ten-dimensional supergravity theories. We find that the AdS solutions, the Hpp-waves and the flat space solutions exhaust them.
Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.
problem Evolution of hypersurfaces in spacetime.
method Weak solutions for hypersurfaces evolving along inverse spacetime mean curvature in asymptotically flat maximal initial data sets.
result Weak solution detects both future- and past-trapped apparent horizons.
We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.
Study gap phenomenon in flat manifolds with Ricci curvature.
problem Understanding curvature decay in flat manifolds.
method Construct solutions to Yamabe flow and analyze curvature decay.
result If curvature decays quickly, manifold must be flat.
Flat isometric immersions in 3D are developable if they are C1,2/3 regular.
problem Characterizing flat isometric immersions in 3D with Hölder continuity.
method Weak second fundamental form, Gauss-Codazzi-Mainardi equations, and degenerate Monge-Ampère equation analysis.
result Isometric immersions of local C1,α regularity with α>2/3 are developable. The paper constructs flat metrics on orbifolds and resolutions.
problem Finding flat metrics on orbifolds and their resolutions.
method Gluing construction and analysis of singularities.
result All crepant resolutions of non-Kähler Calabi-Yau orbifolds with Chern-Ricci flat balanced metrics admit such metrics.
The notion of maximal extension of a globally hyperbolic space-time arises from the notion of maximal solutions of the Cauchy problem associated to the Einstein's equations of general relativity. In 1969 Choquet-Bruhat and Geroch proved that if the Cauchy problem has a local solution, this solution has a unique maximal…