Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.
problem Linear perturbations of Spin(7) metrics.
method Applying the method of linear perturbations to Spin(7)-structures.
result Only rank one nilpotent matrices determine nontrivial perturbations.
Study metric perturbations to make degenerate harmonic forms non-degenerate.
problem Dealing with degenerate harmonic 1-forms in Riemannian geometry.
method Combining analysis of local expansions with Nash-Moser implicit function theorem.
result Proves deformation to nearby non-degenerate Z/2-harmonic 1-forms.
The paper constructs new bimetric conformal invariants using metric perturbations.
problem Developing new conformal invariants in Riemannian geometry.
method Using linear metric perturbations and conformal invariants.
result New bimetric conformal invariants on 4D manifolds are derived.
New metrics found for product spaces with slight changes.
problem Finding metrics for Laplace eigenvalues in perturbed conformal classes.
method Proved existence of extremal metrics for Laplace eigenvalues in perturbed product conformal classes.
result Existence of metrics extremal for some Laplace eigenvalues in perturbed product conformal classes.
The paper shows how to stabilize perturbed Kähler-Ricci solitons.
problem Stabilizing perturbed Kähler-Ricci solitons.
method Normalized Kähler-Ricci flow starting from perturbed metrics.
result The flow converges to an asymptotically conical gradient expanding Kähler-Ricci soliton.
Stability of submanifold cut loci under metric perturbations proved.
problem Stability of submanifold cut loci under metric perturbations.
method Continuity of injectivity radius and Whitney C2 perturbation of submanifolds. result Hausdorff stability of submanifold cut loci under C2 metric perturbations. Paper finds positive metric entropy in perturbed geodesic flow.
problem Understanding dynamics outside KAM tori in nearly integrable systems.
method Lagrangian perturbation of geodesic flow on a flat 3-torus.
result Positive metric entropy found outside some KAM tori.
Paper studies Riesz transform stability under metric perturbations.
problem Stability of Riesz transform boundedness under metric perturbations.
method Derives conditions for stability of Lp-boundedness of Riesz transform. result Provides counter-examples for instability of Riesz transform boundedness.
The paper proves conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
problem Conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
method Study of constant scalar curvature Kähler (cscK) metrics on complete non-compact Kähler--Einstein manifolds.
result Sufficient conditions for a cscK perturbation of a Kähler--Einstein metric to remain Kähler--Einstein.
Extremal metrics found for Laplace eigenvalues in nearby conformal classes.
problem Finding extremal metrics for Laplace eigenvalues in perturbed conformal classes.
method Existence of extremal metrics in a conformal class allows finding similar metrics in nearby classes.
result Perturbed harmonic maps with constant density obtained as part of the arguments.
Study the Dirac operator on a 3-sphere under metric perturbations.
problem Analyze the behavior of eigenvalues of the Dirac operator on a 3-sphere under metric perturbations.
method Derive explicit perturbation formulae for the two eigenvalues closest to zero, considering second variations.
result The eigenvalues closest to zero remain double eigenvalues and are completely determined by the increment of Riemannian volume.
Ricci flow simulations show unstable Fubini-Study metrics develop singularities.
problem Understanding the behavior of unstable perturbations in Ricci flow.
method Numerical simulations of Ricci flow starting from unstable Fubini-Study metrics.
result Ricci flow solutions from unstable Fubini-Study metrics develop local singularities.
New metrics improve scRNA-seq perturbation modeling by reducing mode collapse.
problem Outperformed by simple mean prediction in scRNA-seq perturbation modeling.
method Introduce DEG-aware metrics (WMSE, Rw2(Δ)) and negative/positive baselines. result WMSE loss function reduces mode collapse and improves model performance.
We developed a perturbation model for affine gravity theories.
problem Cosmological perturbations in theories without metric.
method Segregated perturbations into symmetric and antisymmetric components, decomposing into irreducible elements.
result Fully addressed gauge freedom in affine gravity theories.
New Einstein metrics created by modifying hyperbolic infinity.
problem Creating new Einstein metrics.
method Perturbing conformal infinity of geometrically finite hyperbolic metrics and applying inverse function theorem.
result Construct new examples of Einstein metrics.
Estimates mass of static vacuum metrics with small Bartnik data.
problem Estimating mass of static vacuum metrics with small perturbations.
method Second-order mass estimation using Bartnik data.
result New upper bound on Bartnik mass to fifth order.
Researchers derive expressions for metric perturbations of extremal surfaces.
problem Understanding changes in extremal surfaces under metric perturbations.
method Derived explicit expressions for position and surface area changes.
result Found an expansion of surface area involving multiple integrals of geometric quantities.
Stability of cut locus under metric perturbations in compact Riemannian manifolds.
problem Stability of cut locus under C2-perturbations of the metric. method Proving stability with respect to the Hausdorff metric of the cut locus under C2 perturbation of the metric. result The Hausdorff distance between cut loci converges to zero as the metrics converge.
Prove that collapsing CSC metrics can be perturbed to invariant collapsing CSC metrics.
problem Prove that collapsing constant scalar curvature metrics can be perturbed to invariant collapsing constant scalar curvature metrics.
method Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
result Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
Study magnetic perturbations in Riemannian and Lorentzian Calderón problems.
problem Determining metrics from boundary measurements under magnetic perturbations.
method Runge approximation for Riemannian case, microlocal analysis for Lorentzian case.
result Metrics can be uniquely determined in both Riemannian and Lorentzian cases under specific perturbations.
This paper introduces metrics to evaluate robustness of neural networks to natural adversarial examples.
problem Measuring robustness of neural networks to natural adversarial examples.
method Proposes latent space performance metrics based on generative models.
result Latent adversarial perturbations are often perceptually small and associated with classifier accuracy.
We prove a theorem which asserts that the Lie algebra of all holomorphic vector fields on a compact Kähler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal Kähler metric proved by Calabi.
Estimates for harmonic forms on a 3-Torus, proving their existence.
problem Existence of nowhere vanishing harmonic 1-forms on a 3-Torus.
method Explicit computation of injectivity estimates using the Laplace operator on the 3-Torus and its perturbations.
result Existence of a nowhere vanishing harmonic 1-form on a perturbed metric on the 3-Torus.
Complete Calabi-Yau metrics made on special 3D spaces.
problem Creating complete Calabi-Yau metrics on complex 3D spaces.
method Used gluing construction and perturbation argument.
result Produced complete Calabi-Yau metrics with unbounded curvature.
Study reveals class-dependent effects in perturbation-based feature attribution metrics for time series classification.
problem Varying effectiveness of perturbation-based metrics across different classes in time series models.
method Systematic empirical analysis across multiple datasets, model architectures, and perturbation strategies.
result Perturbation-based metrics show varying effectiveness across classes, with some metrics performing better for certain classes.
We construct continuous families of scattering manifolds with the same scattering phase. The manifolds are compactly supported metric perturbations of Euclidean Rn for n≥8. The metric perturbation may have arbitrarily small support.
Uniform K-stability implies Kähler-Einstein metric for Q-Fano varieties.
problem Proving the existence of Kähler-Einstein metrics for uniformly K-stable singular Fano varieties.
method Modified Berman-Boucksom-Jonsson's strategy with perturbative arguments and non-Archimedean estimates.
result Uniform K-stability is equivalent to the existence of Kähler-Einstein metrics for Q-Fano varieties.
We show that a small perturbation of the boundary distance function of a simple Finsler metric on the n-disc is also the boundary distance function of some Finsler metric. (Simple metric form an open class containing all flat metrics.) The lens map is map that sends the exit vector to the entry vector as a geodesic c…
3D spherical caps are rigid under certain perturbations.
problem Rigidity of 3D spherical caps under specific perturbations.
method Gromov's μ-bubble technique
result 3D spherical caps are rigid under perturbations that maintain metric, scalar curvature, and mean curvature.
The study proves the stability of smooth embeddings of Riemannian metrics into Euclidean space.
problem Stability of smooth embeddings of Riemannian metrics into Euclidean space.
method Local perturbation method to derive a time-dependent local perturbation method.
result Construction of a smooth parametrized family of isometric embeddings for a short time.
Generalizes Zermelo navigation problem on Riemannian manifolds with strong perturbation.
problem Navigating on Riemannian manifolds with speed constraints and perturbations.
method Applying Finsler metric of Kropina type to solve the problem.
result Generalized solution to Zermelo navigation problem on Riemannian manifolds with strong perturbations.
Optimizes search paths in river environments using Finslerian geometry.
problem Optimizing search paths in river environments.
method Using Finslerian geometry and time-optimal paths based on Randers metric.
result Time-optimal paths in river environments.
Tseytlin has recently proposed that an action functional exists whose gradient generates to all orders in perturbation theory the Renormalization Group (RG) flow of the target space metric in the worldsheet sigma model. The gradient is defined with respect to a metric on the space of coupling constants which is explici…
The Dirac operator changes smoothly with slight metric tweaks.
problem Smooth dependence of the Dirac operator on metric perturbations.
method Riesz continuity, harmonic analysis, Calderón's commutator, Kato square root problem.
result The Dirac operator is Riesz continuous under L∞ perturbations of metrics. Paper shows perturbed Taub-Bolt metric becomes singularity under Ricci flow.
problem Analyzing stability of Taub-Bolt metric under Ricci flow.
method Box argument and construction of Ricci flows on compact manifolds.
result Compact perturbation of Taub-Bolt metric evolves into finite time singularity.
Paper proposes a robust metric learning algorithm.
problem Robustness of metric learning against adversarial perturbations is insufficient.
method Proposes a novel Mahalanobis distance metric learning algorithm.
result Certifiable robustness improvement over Euclidean distance.
We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.
Study shows current metrics for audio adversarial examples are unreliable for human perception.
problem The reliability of metrics for evaluating audio adversarial examples.
method Analytical framework and human evaluation experiment.
result Current metrics for audio adversarial examples are not reliable for human perception.
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.
We prove the equivalences of several classical complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new Kähler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete Kähler metric with bounded negative holomorphic s…
Proposes a text perturbation method using a Mahalanobis metric to balance privacy and utility.
problem Low utility of text analysis when using spherical noise for privacy-preserving text embedding.
method Regularized Mahalanobis metric to add elliptical noise, accounting for embedding space density.
result Improves privacy statistics while maintaining utility, outperforming Laplace mechanism.
The study shows instability of Nikodym maximal function bounds on Riemannian manifolds under metric perturbation.
problem Instability of Nikodym maximal function bounds on Riemannian manifolds under metric perturbation.
method Analyzing the instability of $L^{ rac{d+2}2}$ bounds for the Nikodym maximal function over manifolds of constant sectional curvature and extending to any d-dimensional Riemannian manifold with a local totally geodesic submanifold. result The instability of the bounds for the Nikodym maximal function on Riemannian manifolds under metric perturbation.
Einstein manifolds are rigid under certain metric deformations.
problem Characterizing Einstein manifolds that resist volume-preserving metric deformations.
method Various characterizations and constructions of mass-decreasing perturbations.
result Constructs mass-decreasing perturbations of specific metrics.
Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigen…
We study the perturbations of two classes of static black ellipsoid solutions of four dimensional vacuum Einstein equations. Such solutions are described by generic off--diagonal metrics which are generated by anholonomic transforms of diagonal metrics. The analysis is performed in the approximation of small eccentrici…
Among all conformal classes of Riemannian metrics on CP2, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the Seiberg-Witten equations, also yields new results on the total scalar curvature of almost-Kähler 4-manifolds.
In this paper we prove the Penrose inequality for metrics that are small perturbations of the Schwarzschild anti-de Sitter metrics of positive mass. We use the existence of a global foliation by weakly stable constant mean curvature spheres and the monotonicity of the Hawking mass.
Counts minimal tori in Riemannian manifolds with 6 or more dimensions.
problem Counting minimal tori in Riemannian manifolds.
method Introduces a function to count minimal tori and shows invariance under metric perturbations.
result The count function is invariant under metric perturbations.