arXiv research

Global inverse function theorem proved easily using Riemannian geometry.

Global implicit function theorem for Fréchet spaces, solving derivative loss problems.

Global Morse index theorem applied to Jacobi fields on CMC surfaces.

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 …

We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming $b_2\geq 5$) which …

Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.

Synthetic splitting theorem for Lorentzian spaces with non-negative curvature.

Proves a synthetic Lorentzian Cartan-Hadamard theorem.

We provide sufficient conditions for the existence of a global diffeomorphism between tame Fréchet spaces. We prove a version of the Mountain Pass Theorem which is a key ingredient in the proof of the main theorem.

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.

Introduces Alexandrov spaces with curvature below, covering various theorems.

Smooth distributions on subcartesian spaces can be globally finitely generated.

The abstract proves a global splitting theorem for Poisson manifolds.

In this paper, the pinching problems of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$ are studied. By making use of the Sobolev inequality, we prove a global pinching theorem of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$.

Paper proves new theorems about curvature in weighted manifolds.

Global homotopies upgrade classical map in differential geometry.

We give sufficient conditions for a $C^1_c$-local diffeomorphism between Fréchet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fréchet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a functio…

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…

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…

Low regularity spacetimes split into simpler structures.

Establishes a Lorentzian Lasry-Lions regularization theorem for functions on globally hyperbolic spacetimes.

We obtain global extensions of the celebrated Nash-Kuiper theorem for $C^{1,θ}$ isometric immersions of compact manifolds with optimal Hölder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent $θ<1…

The paper proves a smooth Birman-Hilden theorem for hyperkähler 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…

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 $\R \times S$.

This paper generalizes Batchelor's theorem in $C^\infty$-superschemes.

The index theorem connects anomalies on a domain wall to global integrals.

We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgro…

Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.

Extended Vaisman theorem 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.

We prove several Liouville-type non-existence theorems for higher order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. These results will be obtained by using theorems of the connections between the geometry of a complete smooth manifold and the global behavio…

In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.

Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism $f: M \to\mathbb{R}^n$ is bijective if and only if $H_{n-1}(M)=0$ and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometr…

We continue the development of $\mathbb{Z}^n_2$-supergeometry, a natural generalization of classical ($\mathbb{Z}_2$-graded) supergeometry, by proving the Frobenius theorem for integrable distributions on differentiable $\mathbb{Z}^n_2$-supermanifolds. Both the local and global versions of the theorem are addressed.

Splitting theorem for non-positively curved Lorentzian spaces.

The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.

Global rigidity theorem for certain lattice actions on manifolds.

The paper proves the existence of area-minimizing hypersurfaces in AF manifolds of higher dimensions.

Global existence and convergence of heat flow for p-harmonic maps.

Paper proves a quantitative estimate for transforming almost complex structures into standard ones.

New Alexandrov-Patchwork construction for Lorentzian spaces with curvature bounds.

Proves Gannon-Lee theorem for $C^1$ spacetimes.