New method calculates discrete curvature using effective resistances.
problem Calculating discrete curvature on graphs.
method Effective resistances to calculate curvature on graph nodes and links.
result Relation to established discrete curvatures and convergence to continuous curvature.
New curvature defined via graph resistances leads to Ricci flow.
problem Defining curvature on graph edges for analysis.
method Introducing Ricci--Foster curvature based on effective resistances and studying Ricci flow.
result Existence of solutions to Ricci flow on short time intervals, preservation of nonnegative curvature.
Proves effective linear volume growth for 3-manifolds with positive scalar curvature.
problem Volume growth of three-manifolds with positive scalar curvature.
method Utilizes the technique of μ-bubbles and almost-splitting theorem.
result Proves effective linear volume growth for 3-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature.
New method uses curvature to improve graph neural networks.
problem Graph Neural Networks struggle with over-smoothing and over-squashing.
method Augmented Forman-Ricci curvature (AFRC) for scalable rewiring.
result AFRC effectively mitigates over-smoothing and over-squashing.
New method proves uniqueness in mean curvature flow.
problem Proving uniqueness in mean curvature flow.
method Arguments from [CM2] to get stronger uniqueness.
result Stronger effective version of uniqueness of blowups.
The study bounds the effective diameter of graphs with positive Ollivier curvature.
problem Bounding the effective diameter of graphs with positive Ollivier curvature.
method Introducing reflective graphs and proving discrete Bonnet Myers theorem.
result The effective diameter bound is attained only for specific graphs.
The paper explores curvature-free effects in manifolds with volume growth and ends-counting.
problem Investigating curvature-free effects in manifolds with volume growth and ends-counting.
method Establishing two main theorems about volume growth and ends-counting.
result Proves the existence of smooth bounded mean-concave exhaustion and escaping geodesic lines.
Study extends compactness theorems to weighted manifolds with integral curvature bounds.
problem Estimating diameter of weighted manifolds under curvature constraints.
method Extended Sprouse's compactness theorems to weighted manifolds with integral curvature bounds. Used ε-range to handle specific cases. Extended segment inequality to weighted manifolds.
result Proved theorems for weighted manifolds with effective dimension ≤ 1 and ≥ dimension.
In this paper, we observe that the brane functional studied in hep-th/9910245 can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three dimensional manifold M has a boundary with strongly positive mean curvature, the effect of this mean curvature can influence…
Local fractional derivatives affect Riemann curvature tensor to zero.
problem Investigating how local fractional derivatives influence the Riemann curvature tensor.
method Introduced a general local fractional derivative operator and defined a specific Riemannian metric tensor field.
result The Riemann curvature tensor of the new metric is identically zero, indicating local isometry to Euclidean space.
The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.
problem Investigating the geometric properties of Riemannian manifolds influenced by average scalar curvature.
method Analyzing the conjugate radius, average area of geodesic spheres, average volume of metric balls, and total volume of closed manifolds.
result Improves the Bishop-Gromov estimate on the average volume of metric balls and proves monotone decreasing properties of certain geometric integrals.
New insights into SGD and generalization via shift-curvature and bias-curvature mechanisms.
problem Understanding the role of curvature in generalization and how SGD affects it.
method Derivation of new SGD steady-state distribution and analysis of shift-curvature and bias-curvature mechanisms.
result Shift-curvature is a significant factor in test performance, especially for small SGD noise.
We point out that a 4-dimensional topological manifold with an Alexandrov metric (of curvature bounded below) and with an effective, isometric action of the circle or the 2-torus is locally smooth. This observation implies that the topological and equivariant classifications of compact, simply connected Riemannian 4-ma…
Paper proposes LCP for structural encodings, outperforming existing methods.
problem Improving Graph Neural Networks performance through effective structural encodings.
method Geometric perspective, Local Curvature Profiles (LCP) for structural encodings, combining with global positional encodings, comparing with rewiring techniques.
result LCP significantly outperforms existing structural encodings and combining LCP with global positional encodings improves performance.
Formula for scalar curvature under metric collapse.
problem Finding positive scalar curvature metrics.
method Formula involving scalar curvature and adapted orthonormal frame.
result Effect of metric collapse on scalar curvature.
The study examines hypersurfaces close to constant mean curvature and their proximity to spheres.
problem Understanding hypersurfaces close to constant mean curvature and their proximity to spheres.
method Quantitative stability results for hypersurfaces with mean curvature close to a constant.
result Hypersurfaces close to constant mean curvature are closely related to spheres, with quantitative descriptions of proximity.
Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.
problem Mean curvature flow of totally real submanifolds.
method Quantitative analysis of almost minimal submanifolds.
result Established convergence result for mean curvature flow.
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…
Study on curvature in finitely generated groups, showing positive curvature in specific cases.
problem Understanding curvature in finitely generated groups.
method Analyzing dead-end elements and related elements to find curvature, studying effect of radius.
result Examples of positive curvature for arbitrary radius in lamplighter and Houghton's group.
New stability analysis improves generalization of multipass SGD.
problem Improper preconditioning affects generalization in multipass SGD.
method Developed on-average stability analysis for multipass SGD.
result Proper preconditioning yields optimal effective dimension dependence.
Conditions for torsion-free connections with specific curvature maps are derived.
problem Finding conditions for torsion-free connections with prescribed curvature.
method Using a power series approach to derive necessary and sufficient conditions for a curvature map to arise from a torsion-free connection.
result A unique torsion-free connection is derived from a given curvature map.
The paper examines curvature and stability in quasi-geostrophic motions using spherical harmonics.
problem Analyzing the curvature and stability of quasi-geostrophic motions.
method Utilizing spherical harmonics and structure constants, the curvature of the L2 metric on the central extension is computed. result A lower bound for weather prediction error in a simplified model is suggested.
In this paper, we prove that nonnegative polyharmonic functions on the upper half space satisfying a conformally invariant nonlinear boundary condition have to be the "\emph{polynomials} plus \emph{bubbles}" form. The nonlinear problem is motivated by the recent studies of boundary GJMS operators and the Q-curvature …
Chirality affects the curvature of molecular networks, influencing their shape and stability.
problem Understanding how chirality influences the curvature of molecular networks.
method Langevin dynamics simulations and constrained gradient optimization of square lattice networks.
result Linking chirality dictates the sign of Gaussian curvature in molecular chainmail networks.
Study on evolving interfaces with complex curvature and density effects.
problem Understanding the dynamics of evolving heterogeneous elastic interfaces.
method Modeling an evolving curve with a density function, analyzing the associated gradient flow evolution.
result Analysis of the preservation and asymptotic behavior of geometric properties in the evolving system.
Paper constructs hyperbolic metrics using circle packings and curvature parameters.
problem Creating polyhedral metrics for surfaces of various topologies.
method Using circle packings and curvature parameters, the paper constructs hyperbolic polyhedral metrics.
result Unified approach to producing polyhedral metrics for surfaces of broader topological types.
We extend the idea and techniques in \cite{Miao} to study variational effect of the boundary geometry on the ADM mass of an asymptotically flat manifold. We show that, for a Lipschitz asymptotically flat metric extension of a bounded Riemannian domain with quasi-convex boundary, if the boundary mean curvature of the ex…
This paper improves HNNs by learning optimal curvature for better generalization.
problem Inappropriate curvatures in HNNs lead to suboptimal performance.
method Sharpness-aware curvature learning method to smooth loss landscape.
result Proposed method improves HNNs' generalization across various settings.
Vacuum gravity shows black holes can form without collapse.
problem Formation of black holes in vacuum gravity.
method Cauchy--double-null framework, Yau's boundary criterion.
result Black holes form from boundary effects in vacuum spacetime.
In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound RicN≥K with K>0 for the negative effective dimension N<0. We analyze two 1-dimensional examples of constant curvature RicN≡K with finite and infinite total volumes.…
Effective Yau-Tian-Donaldson conjecture for spherical varieties.
problem Finding effective K-stability criteria for spherical varieties.
method Formulated an effective variant of the Yau-Tian-Donaldson conjecture and reviewed effective K-stability criteria for spherical varieties.
result Effective K-stability criteria can be computed given combinatorial data.
Paper introduces new flows to find circle packings with specific curvature.
problem Finding circle packings with prescribed total geodesic curvatures.
method Introduces combinatorial Calabi flow, fractional combinatorial Calabi flow, and combinatorial p-th Calabi flow.
result Establishes conditions for the longtime behaviors of these flows.
We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive μ-reach can be estimated by the same curvature measures of the offset of a compact set…
State-of-the-art classifiers have been shown to be largely vulnerable to adversarial perturbations. One of the most effective strategies to improve robustness is adversarial training. In this paper, we investigate the effect of adversarial training on the geometry of the classification landscape and decision boundaries…
Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…
Study local curvature estimates and existence of conformal metrics on noncompact manifolds.
problem Deriving local C0-estimates and existence of conformal metrics with prescribed curvature. method Utilizing Aviles-McOwen's result and its nonlinear extension, combined with asymptotic conditions.
result Proved existence of complete conformal metrics with prescribed curvature functions.
A gap in the proof of the main result in reference [1] in our original submission propagated into the constructions presented in the first version of our manuscript. In this version we give an alternative proof for the existence of Riemannian metrics with positive Ricci curvature on an infinite subfamily of closed, sim…
In this paper, we discuss the invariant properties of curvatures effected by the Matsumoto's C or L-process. We find equivalent conditions on curvatures by comparing the difference between the corresponding curvatures of closely related connections. As an application, Matsumoto's L-process on Randers manifold is …
We use a local argument to prove if an r-dimensional torus acts isometrically and effectively on a connected n-dimensional manifold which has positive kth-intermediate Ricci curvature at some point, then r≤⌊2n+k⌋. This symmetry rank bound generalizes those established by Gr…
We give a relationship that yields an effective geometric way of evaluating mean curvature of surfaces. The approach is reminiscent of the Gauss's contour based evaluation of intrinsic curvature. The presented formula may have a number of potential applications including estimating the normal vector and mean curvature …
Generalization in nonlinear least squares can be studied via algorithmic stability and effective dimension.
problem Generalization in nonlinear least squares models
method Deriving error bounds for local minimizers using algorithmic stability and effective dimension
result Bounds depend on learned geometry rather than parameter count
Study spinorial Yamabe problem on product manifolds, proving spike layer solutions.
problem Existence of solutions on product manifolds with curvature effect.
method Variational methods and blow-up techniques.
result Existence of solutions depending only on one factor and exhibiting spike layers.
Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a sheeting theorem (in all dimensions) for weakly stable CMC hypersurfaces, giving …
We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bounded below by 1 (after possibly rescaling) are macroscopically 1-dimensional on the scales greater than 1. As a consequence we prove that compact manifolds with positive isotropic curvature have virtually free fundamental groups.…
The paper calculates curvature formulas for stable Higgs bundles.
problem Investigating the geometry of stable Higgs bundles over complex manifolds.
method Applying curvature formulas from Schumacher-Toma and Biswas-Schumacher to calculate and analyze the curvature of the base manifold.
result Explicit calculation of holomorphic sectional curvature for a complex Finsler metric.
This paper deals with the concept of curvature of framed space curves, their higher-order derivatives, variations, and co-rotational derivatives. We realize that parametrizing rotation tensor using the Gibbs vector is effective in deriving a closed form formula to obtain any order derivative of the curvature tensor as …
Solves open problems on curved projective varieties.
problem Structure theorems for curved projective varieties.
method Supplements and proposes open problems.
result Provides new insights into structure of curved projective varieties.
We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play …