New principles prove precompactness of domains with lower Ricci curvature bound.
problem Proving precompactness of domains with lower Ricci curvature bound.
method Quantitative Hopf-Rinow theorem and doubling property.
result New precompactness principles applicable to incomplete Riemannian manifolds.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
The Hopf-Rinow theorem is extended to sub-Finslerian geometry.
problem Extending the Hopf-Rinow theorem to sub-Finslerian manifolds.
method Investigation of sub-Finslerian bundle, exponential map, and Legendre transformation.
result Established a relation between completeness, geodesic completeness, and compactness in sub-Finslerian geometry.
Extends Hopf-Rinow theorem to semi-Riemannian spacetimes.
problem Generalizing Hopf-Rinow theorem to compact Lorentzian manifolds.
method Develops null distance for proper cone structures and (n−ν,ν)-spacetimes. result Generalizes Hopf-Rinow theorem to a new class of semi-Riemannian manifolds.
Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.
problem Investigate magnetic geodesics on half-Lie groups using Riemannian and two-form structures.
method Define Mañé's critical value, prove Finsler geodesic flow equivalence, and apply Hopf-Rinow theorem.
result Hopf-Rinow theorem holds for energies above Mañé's critical value on magnetic geodesics.
Geodesic connectedness proved for statistical manifolds with divisible cubic forms.
problem Geodesic connectedness of affine connections on statistical manifolds with divisible cubic forms.
method Analogy with Hopf-Rinow theorem in Riemannian geometry, establishing geodesic completeness.
result Geodesic connectedness established for statistical manifolds with divisible cubic forms.
The paper defines a critical value for a magnetic system and extends solutions beyond blow-up.
problem Analyzing blow-up behavior and extending solutions for a magnetic system.
method Formulated as a magnetic geodesic equation on an infinite-dimensional Lie group, computed Mañé's critical value, established Hopf-Rinow theorem.
result Computed Mañé's critical value for the magnetic two-component Hunter-Saxton system and extended solutions beyond blow-up.
Study on Banach half-Lie groups and their properties.
problem Characterizing and understanding the properties of half-Lie groups in infinite dimensions.
method Investigation of Banach half-Lie groups, their extensions, and right invariant strong Riemannian metrics.
result The full Hopf--Rinow theorem holds for Banach half-Lie groups, a surprising result.
In this paper we will investigate the global properties of complete Hilbert manifolds with upper and lower bounded sectional curvature. We shall prove the Focal Index Lemma that we will allow us to extend some classical results of finite dimensional Riemannian geometry such as Rauch and Berger Theorems and the Topogono…
In this paper, we study the theory of geodesics with respect to the Tanaka-Webster connection in a pseudo-Hermitian manifold, aiming to generalize some comparison results in Riemannian geometry to the case of pseudo-Hermitian geometry. Some Hopf-Rinow type, Cartan-Hadamard type and Bonnet-Myers type results are establi…
After defining generalizations of the notions of covariant derivatives and geodesics from Riemannian geometry for reductive Cartan geometries in general, various results for reductive Cartan geometries analogous to important elementary results from Riemannian geometry are proven using these generalizations. In particul…
A method to generalize results from Riemannian Geometry to Finsler geometry is presented. We use the method to generalize several results that involve only metric conditions. Between them we show that the topology induced by the Finsler structure is equivalent to the manifold topology, we provide a new proof of the Hop…
Develops a lifting theory for exponential maps in semi-Riemannian geometry.
problem Overcoming singularities in exponential maps to prove geodesic connectivity.
method Lifting theory for semi-Riemannian manifolds with path-continuation property.
result General path-lifting theorem extending globally under certain conditions.
Introduce sub-Randers metrics by adding a one-form to a sub-Riemannian metric
problem Define a new class of sub-Finsler metrics
method Derive equations for sub-Randers normal geodesics
result Prove a Hopf-Rinow type theorem for sub-Randers manifolds
Proves globally hyperbolic spacetimes via null distance completeness.
problem No Hopf-Rinow Theorem in Lorentzian Geometry.
method Observation of null distances and their behavior with time functions.
result Proves globally hyperbolic spacetimes via null distance completeness.
Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky g…
In this paper, we give a proof of the quantitative Morse theorem stated by {Y. Yomdin} in \cite{Y1}. The proof is based on the quantitative Sard theorem, the quantitative inverse function theorem and the quantitative Morse lemma.
Paper surveys balanced metrics and proves a geodesic convexity result.
problem Understanding balanced metrics and stability in algebraic geometry.
method Survey and proof of geodesic convexity result.
result Geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive.
The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullb…
Proves quantitative Alexandrov theorem for capillary surfaces.
problem Proving a quantitative version of the Alexandrov theorem for capillary hypersurfaces.
method Quantitative analysis of Montiel-Ros-type argument.
result Generalizes Julin-Niinikoski's result to capillary case.
We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…
Proves theorem for Riemannian manifolds, extending previous work.
problem Proving Quantitative Fatou Theorem on Riemannian manifolds.
method Extending ε-approximation lemma to manifold setting.
result Proves Quantitative Fatou Theorem for Lipschitz domains on Riemannian manifolds.
The study proves a quantitative functional CLT for neural networks with smooth activation functions.
problem Understanding the convergence rates of neural networks with different activation functions.
method Functional versions of the Stein-Malliavin approach and a quantitative functional central limit theorem.
result Rates of convergence depend on the smoothness of the activation function, ranging from logarithmic to sqrt(n).
The paper extends completeness notions to low-regularity spacetimes.
problem Defining completeness conditions for spacetimes with low-regularity metrics.
method Extending Beem's completeness notions to Lorentzian length spaces and proving relationships between them.
result Equivalence of completeness conditions for globally hyperbolic C1-spacetimes under certain conditions. Proves a quantitative index theorem for positive scalar curvature metrics.
problem Studying conjectures and open questions on positive scalar curvature.
method Quantitative relative index theorem and λ-Lipschitz rigidity theorem. result Positive answers to Gromov's open questions on scalar curvature.
We prove a quantitative version of Obata's Theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the genera…
Paper analyzes arbitrage in uncertain markets, providing quantitative asset pricing.
problem Dealing with model uncertainty in markets that allow small arbitrage.
method Quantitative analysis of arbitrage, focusing on asset price processes close to martingales.
result Quantitative version of the Fundamental Theorem of Asset Pricing and Super-Replication Theorem.
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 geodesic Lie groups' convergence to limits with quantitative estimates.
problem Quantifying convergence rates of geodesic Lie groups to their limits.
method Estimates on the difference between original metrics and asymptotic/tangent metrics.
result Sharpens existing bounds on convergence rates.
Quantum neural networks can approximate noisy functions accurately.
problem Approximating noisy functions with quantum neural networks.
method Universal approximation theorem with error bounds for noisy quantum neural networks.
result Quantum neural networks can approximate noisy functions with precise error bounds.
Study connects manifold complexity to scalar curvature bounds.
problem Understanding the relationship between manifold complexity and scalar curvature.
method Combining quantitative operator K-theory, Lipschitz topological K-theory, and a vanishing theorem.
result Established a relationship between covering complexity and scalar curvature bounds.
The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures f(k1,…,kn−1) satisfying suitable conditions. In this paper…
Quantifies the crossing number of knots based on genus and braid index.
problem Estimating the crossing number of knots given their genus and braid index.
method Quantitative Birman-Menasco finiteness theorem applied to crossing numbers.
result Estimates the crossing number of knots in terms of genus and braid index.
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.
Quantifies Schur's theorem for curves in CAT(k) spaces.
problem Quantifying Schur's comparison theorem for curves in CAT(k) spaces.
method Comparison formula for curves in model planes, curvature measures, moment arm, and Reshetnyak's theorem.
result Sharpens and extends classical arm and bow lemmas and Riemannian analogues.
We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let S be a C2 closed embedded hypersurface of Rn+1, n≥1, and denote by osc(H) the oscillation of its mean curvature. We prove that there exists a positive ε, depending on n and upper …
Given a complex structure J on a real (finite or infinite dimensional) Hilbert space H, we study the geometry of the Lagrangian Grassmannian Λ(H) of H, i.e. the set of closed linear subspaces L⊂H such that J(L)=L⊥. The complex unitary group U(HJ), consisting of the elements of the orthogona…
Proves positive mass theorem for hyperbolic manifolds with ends.
problem Establishing positive mass theorem for complex initial data sets.
method Used spectral PSC, Jang equation, and quantitative shielding theorem.
result Proved positive mass theorem for asymptotically hyperbolic manifolds.
We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.
Quantifies closeness of special Lagrangians under Floer conditions.
problem Estimating closeness of special Lagrangians.
method Floer theoretic conditions leading to quantitative estimates.
result Strong-weak uniqueness theorem for special Lagrangians.
We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.
We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some r…
Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.
problem Understanding the p-Laplacian on RCD(K,N) spaces.
method Proving a Talenti-type comparison theorem.
result Sharp, rigid and stable Talenti-type comparison theorem.
The paper proves a quantitative positive mass theorem for spin manifolds with distance estimates.
problem Proving a positive mass theorem for spin manifolds with arbitrary ends.
method Analyzing the scalar curvature and using distance estimates.
result Quantitative answer to Schoen and Yau's question on the positive mass theorem.
In this article we extend to generic p-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case p=2. We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of…
This paper studies neural network operators and their convergence properties.
problem Understanding the approximation and convergence of neural network operators.
method Proves density results, convergence estimates, and Voronovskaya-type theorems.
result Establishes quantitative convergence estimates and derives Voronovskaya-type theorems.
Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.
problem Quantifying rigidity in Alexandrov spaces with curvature constraints.
method Using Gromov-Hausdorff distance and properties of Alexandrov spaces.
result Alexandrov spaces with curvature bounds are close to hyperbolic manifolds.
This paper proves a new, more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.
problem Proving a more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.
method Using a combination of Ricci curvature bounds and Riemannian universal cover properties to establish a quantitative rigidity result.
result If a manifold has positive Ricci curvature and a diameter close to the maximal possible, it is diffeomorphic and bi-Hölder close to the sphere.