Invites into Alexandrov geometry with applications and theorems.
problem None explicitly stated, focuses on Alexandrov geometry beauty and power.
method Various Alexandrov geometry theorems and constructions.
result Explains the power and applications of Alexandrov geometry.
Novel geodesic results on affine and Lorentzian manifolds.
problem Existence and multiplicity of geodesics on manifolds.
method Path-lifting and path-continuation properties of exponential maps.
result Generalization of Hadamard-Cartan theorem to affine manifolds.
These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian and semi-Riemannian structures on manifolds. Affine connections, geodesics, tors…
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.
Global implicit function theorem for Fréchet spaces, solving derivative loss problems.
problem Solving initial value problems with derivative loss in Fréchet spaces.
method Global implicit function theorems for Keller's Cc1-mappings in Fréchet spaces, applied through submersions and transversality. result Global existence and uniqueness of solutions to initial value problems with derivative loss.
Global diffeomorphism proof between tame Fréchet spaces.
problem Existence of global diffeomorphism between Fréchet spaces.
method Mountain Pass Theorem and sufficient conditions for diffeomorphism.
result Existence of global diffeomorphism between tame Fréchet spaces.
Global moduli theory for symplectic varieties proven.
problem Proving global moduli theory for symplectic varieties.
method Developed global moduli theory following Beauville's approach.
result Proved a global Torelli theorem for symplectic varieties.
Global diffeomorphism theorem for Fréchet spaces established.
problem Establishing global diffeomorphisms in Fréchet spaces.
method Extending local diffeomorphisms to global ones, using generalized gradients and Lipschitz functions.
result Global diffeomorphism theorem proved for Fréchet spaces.
Global Morse index theorem applied to Jacobi fields on CMC surfaces.
problem Existence and structural theorem of Jacobi fields on CMC surfaces.
method Global Morse index theorem proof via set-continuity of domain shapes and eigenvalue continuity.
result Global Morse index theorem provides structural existence of Jacobi fields.
Compact hyperkaehler manifolds are higher-dimensional generalizations of K3 surfaces. The classical Global Torelli theorem for K3 surfaces, however, does not hold in higher dimensions. More precisely, a compact hyperkaehler manifold is in general not determined by its natural weight-two Hodge structure. The text gives …
Global Nash-Kuiper theorem extended for compact manifolds with optimal Hölder exponent.
problem Isometric immersions of compact manifolds with optimal Hölder exponent.
method Global extensions of Nash-Kuiper theorem for C1,θ isometric immersions. result Nash-Kuiper non-rigidity prevails up to exponent θ<1/5 for isometric embeddings of convex compact surfaces. Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.
problem Synthetic proof of rigidity for globally hyperbolic Lorentzian spaces.
method Synthetic geometry and warped product analysis.
result Spaces with specific curvature and distance realizer are warped products.
The study proves non-existence theorems for Codazzi tensors on Riemannian manifolds.
problem Proving non-existence theorems for Codazzi tensors on Riemannian manifolds.
method Using theorems connecting manifold geometry and subharmonic functions.
result Several Liouville-type non-existence theorems for Codazzi tensors.
Synthetic splitting theorem for Lorentzian spaces with non-negative curvature.
problem Proving a splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature.
method Synthetic approach using triangle comparison and parallelity of timelike lines.
result Establishes a splitting of a neighborhood of a complete timelike line, leading to global inextendibility.
Proves a synthetic Lorentzian Cartan-Hadamard theorem.
problem Formulates and proves a theorem for Lorentzian geometry.
method Uses an appropriate notion of local concavity for Lorentzian (pre-)length spaces.
result Establishes existence and uniqueness of timelike geodesics.
Introduces Alexandrov spaces with curvature below, covering various theorems.
problem Understanding spaces with curvature constraints.
method Explains comparison conditions, globalization, tangent spaces, etc.
result Globalization theorem and other theorems established for Alexandrov spaces.
Smooth distributions on subcartesian spaces can be globally finitely generated.
problem Understanding smooth distributions on subcartesian spaces.
method Embedding in Euclidean space, Whitney Embedding Theorem, and distribution theory.
result Smooth generalized distributions and subbundles on connected subcartesian spaces are globally finitely generated.
We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.
The null splitting theorem (proved in math.DG/9909158) is discussed. As an application, a uniqueness theorem for Minkowski space and for de Sitter space associated with the occurrence of null lines (inextendible globally achronal null geodesics) is presented.
We prove some Bernstein theorems for entire space-like submanifolds in pseudo-Euclidean spaces and, as a corollary, we obtain a new proof of the Calabi-Pogorelov theorem on global solutions of Monge-Ampere equations.
The abstract proves a global splitting theorem for Poisson manifolds.
problem Decomposing compact Kähler Poisson manifolds into simpler components.
method Proving a global splitting theorem using symplectic leaves and finite étale covers.
result Compact Kähler Poisson manifolds can be split into simpler components.
Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
In this paper, the pinching problems of complete λ-hypersurfaces in a Euclidean space Rn+1 are studied. By making use of the Sobolev inequality, we prove a global pinching theorem of complete λ-hypersurfaces in a Euclidean space Rn+1.
Global homotopies upgrade classical map in differential geometry.
problem Upgrade classical Hochschild-Kostant-Rosenberg map to a deformation retract.
method Combining symbol calculus and coalgebraic van Est theorem.
result Develop deformation retracts in various settings.
The study introduces a new function to analyze special holonomy manifolds.
problem Analyzing harmonic forms on special holonomy manifolds.
method Introducing a global perturbation potential function and studying its effects on harmonic forms.
result Established vanishing theorems on L2 harmonic forms under certain conditions. Local to global arguments on low dimensional manifolds simplify complex structures.
problem Proving a deep theorem of Thurston in foliation theory.
method Using contractibility of complexes associated to submanifolds to cut manifolds into simpler pieces.
result Different proof of Thurston's theorem without foliation theory.
The study provides structure theorems for singular minimal laminations.
problem Understanding the structure of singular minimal laminations in 3D space.
method Local removable singularity theorem and local picture theorem applied to topology.
result Global structure theorems for certain singular minimal laminations.
We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk D that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeom…
The local-to-global property is proven for Morse quasi-geodesics in various groups.
problem Proving local-to-global properties for Morse quasi-geodesics in different groups.
method Developing a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces.
result Generalization of combination theorems for stable subgroups of various groups.
New lemma extends global correspondence to all dimensions, proving new theorems.
problem Global correspondence between hypersurfaces and conformal metrics in hyperbolic space.
method Developed new lemma about conformal factors' asymptotic behavior.
result Global correspondence and embeddedness theorems extended to all dimensions.
We extend Banach's fixed point theorem to establish convergence of iterative methods under carefully chosen metrics.
problem Limited applicability of Banach's fixed point theorem in non-convex problems.
method Developed a strong converse theorem to Banach's fixed point theorem, showing its universal applicability for convergence of iterative methods.
result We show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and can be used to bound the convergence rate.
The paper proves a global geometric formula for volume holonomy in gauge theory.
problem Describing higher parallel transport in classical principal bundle theory.
method Global geometric approach to parallel transport on surfaces and volumes.
result Global formula for volume holonomy and gauge invariance.
Low regularity spacetimes split into simpler structures.
problem Proving splitting theorem for C1 metrics and weights. method Combining elliptic techniques and line-adapted curves.
result Extends Lorentzian splitting theorem to C1 settings. We introduce a notion of probabilistic convexity and generalize some classical globalization theorems in Alexandrov geometry. A weighted Alexandrov's lemma is developed as a basic tool.
We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…
Geometric proof for averaging theorem on Riemannian manifolds.
problem Averaging theorem for perturbed dynamical systems on Riemannian manifolds.
method Geometric proof using a free coordinate approach.
result Generalization of averaging procedure to any open domain with compact closure.
Establishes a Lorentzian Lasry-Lions regularization theorem for functions on globally hyperbolic spacetimes.
problem Optimal transport with C1,1 regularizing pairs method Local semiconcavity and future-directed timelike superdifferentials
result Derives C1,1 regularizing pairs for optimal transport under general assumptions The paper proves a smooth Birman-Hilden theorem for hyperkähler manifolds.
problem Understanding the smooth mapping class groups of certain 4-manifolds.
method Geometric and Teichmüller-theoretic methods.
result Proves a global Torelli theorem for generalized Enriques manifolds.
The Gannon-Lee singularity theorems give well-known restrictions on the spatial topology of singularity-free (i.e., nonspacelike geodesically complete), globally hyperbolic spacetimes. In this paper, we revisit these classic results in the light of recent developments, especially the failure in higher dimensions of a c…
This paper generalizes Batchelor's theorem in C∞-superschemes.
problem Establishing global splittings in C∞-superschemes. method Structural generalization of Batchelor's theorem.
result Global splittings of C∞-superspaces can be characterized by Euler vector fields. The index theorem connects anomalies on a domain wall to global integrals.
problem Relating anomalies on a domain wall to global integrals.
method Formulated and proved an analog of the Atiyah-Patodi-Singer theorem.
result The index is expressed through global chiral and parity anomalies.
Given a globally hyperbolic spacetime M, we show the existence of a {\em smooth spacelike} Cauchy hypersurface S and, thus, a global diffeomorphism between M and R×S.
Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.
problem Understand the global structure of spacetimes with weakly trapped surfaces.
method Show foliation of MOTS generating totally geodesic null hypersurfaces.
result Obtain local or global rigidity results based on assumptions.
Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
A multiplicatively closed, horizontal foliation on a Lie groupoid may be viewed as a "pseudoaction" on the base manifold M. A pseudoaction generates a pseudogroup of transformations of M in the same way an ordinary Lie group action generates a transformation group. Infinitesimalizing a pseudoaction, one obtains the…
Local gap theorem for Ricci shrinkers ensures flatness if certain functionals are close to zero.
problem Understanding the global geometry of Ricci shrinkers from local information.
method Proving a local gap theorem using the local μ-functional. result Ricci shrinkers are flat if the local μ-functional is close to zero. The paper explores curvature and minimal surfaces in normed spaces.
problem Defining and studying curvature and minimal surfaces in normed spaces.
method Characterizing minimal surfaces and proving global theorems.
result Several characterizations of minimal surfaces and analogues of global theorems are derived.
Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.
problem Eigenvalue formulas and rigidity theorems for weighted manifolds.
method Extends variational formulae to weighted manifolds, proving global rigidity theorems.
result Global rigidity theorems for critical domains in Gaussian half-space.