Study integral bounds for submanifolds in low codimension with topological implications.
problem Integral curvature bounds and topological obstructions for submanifolds.
method Integral curvature bounds in terms of Betti numbers, δ-pinched immersions, and pinched second fundamental form. result Obtained topological obstructions for δ-pinched immersions and intrinsic obstructions for minimal submanifolds in spheres. The paper extends spacetime topology results using codimension 2 null cut locus properties.
problem Understanding spacetime topology with and without horizons.
method Review and extension of existing literature on spacetime topology, utilizing codimension 2 null cut locus properties.
result Results for spacetimes with and without horizons, including asymptotically AdS settings.
Connected sums defined for codimension two locally flat submanifolds in higher dimensions.
problem Defining connected sums for codimension two locally flat submanifolds in various dimensions.
method Using results from higher dimensional topological manifolds and four-manifolds, defining connected sums for codimension two locally flat submanifolds.
result A well-defined connected sum exists up to orientation preserving homeomorphism.
Researchers prove an L-theoretic signature transfer in codimension 2.
problem Proving an L-theoretic signature invariance in codimension 2. method Constructing a transfer map between symmetric L-groups. result Signature transfer up to a torsion of order at most 4.
The standard P. A. Smith theory of p-group actions on spheres, disks, and euclidean spaces is extended to the case of p-group actions on tori (i.e., products of circles) and coupled with topological surgery theory to give a complete topological classification, valid in all dimensions, of the locally linear, orientation…
Summing submanifolds in topological spaces is possible and explicit methods are provided.
problem Summing submanifolds in topological spaces.
method Explicit construction of submanifolds representing the sum of homology classes.
result Explicit construction of submanifolds representing the sum of homology classes.
Enhances Molino's description for foliated spaces, simplifying their study.
problem Complexity in studying foliated spaces, especially in non-trivial cases.
method Introduces a compact topological group action and C∞ version, characterizing foliated homogeneity. result Characterizes compact minimal G-foliated spaces and their foliated homogeneity. In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded belo…
Research explores real algebraic realization of round fold maps of codimension -1.
problem Real algebraic realization of round fold maps of codimension -1.
method Generalizes canonical projections of unit spheres to round fold maps and discusses their real algebraic realization.
result Developed new studies in real algebraic geometry focusing on round fold maps of codimension -1.
Identifies submanifolds as topological spheres in hyperbolic space.
problem Characterizing submanifolds as topological spheres in hyperbolic space.
method Defines conditions on Ricci curvature and mean curvature vector length.
result Identifies submanifolds as topological spheres under given conditions.
Research describes all possible gradient vector fields on a sphere with up to ten singular points.
problem Characterizing gradient vector fields on a sphere with limited singular points.
method Using a graph to represent one-dimensional stable manifolds, specifying singularities and connections.
result Identified all topological structures of codimension one gradient vector fields on a sphere with up to ten singular points.
We show that every volume preserving codimension one Anosov flow on a closed Riemannian manifold of dimension greater than three admits a global cross section and is therefore topologically conjugate to a suspension of a linear toral automorphism. This proves a conjecture of Verjovsky from the 1970's in the volume pres…
New findings show exotic non-leaves in 4-manifolds with infinitely many ends.
problem Understanding non-leaves in smooth manifolds with low regularity.
method Developed new criteria for non-leaves in C1,0 and C2 regularity. result Found exotic non-leaves in S4 punctured by a Cantor set. Builds torus fibrations over singular manifolds with specific features.
problem Creating torus fibrations over manifolds with singularities.
method Constructs a topological space X as a torus fibration over an integral affine manifold B with singularities. result The fibration has a discriminant in codimension 2.
We provide integral curvature bounds for compact Riemannian manifolds that allow isometric immersions into a Euclidean space with low codimension in terms of the Betti numbers.
New method for high-dimensional submanifolds using surgery and curvature control.
problem Mean curvature flow in high codimension with topological control.
method Mean curvature flow with surgery, new a priori estimates for second fundamental form.
result Sharp classification of quadratically 2-convex submanifolds in higher codimensions.
Study Seiberg-Witten theory on manifolds with codimension-3 foliations.
problem Analyzing Seiberg-Witten theory on manifolds with codimension-3 Riemannian foliations.
method Construct basic Seiberg-Witten invariant and monopole Floer homologies for each transverse spin structure.
result Monopole Floer homologies are independent of the bundle-like metric and generic perturbation.
Study on fractional mass for codimension-two currents, proving equi-coercivity and Γ-convergence.
problem Defining and studying fractional mass for codimension-two currents on manifolds.
method Energy minimization with Jacobian constraint, equi-coercivity, Γ-convergence, weak linking.
result Equivalence of two formulations of fractional mass, improved regularity for s-harmonic maps. The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity, global existence and convergence of the flow in various ambient spaces and codimensio…
Study topological invariants for hypersurfaces using vector fields.
problem Finding obstructions to foliations on hypersurfaces.
method Define a perturbed Gauss map to derive topological invariants.
result Obtained topological invariants that depend on manifold geometry.
Study on stable translating solitons without complex cycles.
problem Understanding the topology of stable translating solitons.
method Analyzing complete f-stable translating solitons to show absence of specific cycles. result Two-dimensional complete f-stable translating solitons have genus zero. Minimal hyperbolic foliations on 3-manifolds have non-simply connected generic leaves.
problem Characterizing surfaces with minimal hyperbolic foliations.
method Analyzing codimension one foliations on closed 3-manifolds.
result Noncompact surfaces satisfying a specific condition are homeomorphic to the leaf of a minimal foliation with non-simply connected generic leaf.
Study on instantons over product manifolds with a codimension-4 form.
problem Characterize dimension reduction for moduli spaces of generalized ASD instantons.
method Integrability results for families of connections, topological criteria, and explicit descriptions of moduli spaces.
result Complete characterization of dimension reduction for moduli spaces of generalized ASD instantons.
The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In …
We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding is at least two. This is a topological counterpart of the algebraic splitting theorem for embedded cobordisms of the first author, A. Nemethi…
The paper studies affine manifolds with linear foliations and their topological properties.
problem Characterizing the topology of affine manifolds with linear foliations.
method Analyzing the structure of affine manifolds and their foliations, proving topological properties.
result Compact affine manifolds with linear foliations are homeomorphic to the torus under certain conditions.
We study the topology of the space of smooth codimension one foliations on a closed 3-manifold. We regard this space as the space of integrable plane fields included in the space of all smooth plane fields. It has been known since the late 60's that every plane field can be deformed continuously to an integrable one, s…
Study introduces fractional mass concept for surfaces, proving its convergence.
problem Understanding fractional mass on surfaces.
method Introduces fractional s-mass, proves Γ-convergence and pointwise convergence. result Fractional s-mass converges to (n−2)-dimensional area. Survey on geometric, analytic, and topological aspects of 4D equations.
problem No specific problem stated in abstract.
method Geometric, analytic, and topological discussions.
result New solution of the Cauchy problem over null hypersurfaces.
We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{ö}nflies theorem. As an application, we show that one can read off oriented matroids from arrangement…
We provide a strengthening of Jordan separation, to the setting of maps from a compact topological space X into a sphere, where the source space X is not necessarily a codimension one sphere, and the map is not necessarily injective.
New proof shows no equifocal submanifolds with non-flat sections in certain symmetric spaces.
problem Existence of equifocal submanifolds with non-flat sections in symmetric spaces.
method Introduced slice topology, universal covering, and topological Tits buildings to derive contradiction.
result No equifocal submanifolds with non-flat sections in certain symmetric spaces.
Let f:M→M be an expansive homeomorphism with dense topologically hyperbolic periodic points, M a compact manifold. Then there is a local product structure in an open and dense subset of M. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is …
Generic singularities found in spacetimes with weakly trapped submanifolds.
problem Existence of singularities in spacetimes with weakly trapped submanifolds.
method Using Whitney topologies and singularity theorems, the study shows genericity of singularities in spacetimes with weakly trapped submanifolds.
result Generic singularities are found in spacetimes with weakly trapped submanifolds.
Author defines products of elements in cobordism-like modules induced from generic maps.
problem Generalizing cobordism modules for negative codimension generic maps.
method Defining products for pairs of elements in cobordism-like modules.
result Suitable elements for products defined in cobordism-like modules.
Study on compact hypersurfaces in spheres with Ricci curvature bounds.
problem Topology of compact hypersurfaces in spheres with Ricci curvature constraints.
method Use of Bochner technique for stronger results.
result Stronger results than previous studies in higher codimensions.
Defines a map from hypersurfaces to spheres using vector fields, linking curvature and topological invariants.
problem Obstructing codimension one foliations on closed hypersurfaces.
method Defines a Gauss map using a unit vector field and uses it to create topological invariants.
result Shows these invariants can be used as obstructions to foliations.
We study a moduli stratum of A-orbits of plane-to-plane germs of corank 2 with codimension 3. We describe explicitly the bifurcation diagram of its topologically A-versal unfolding. Two geometric applications to parabolic objects are presented.
Study cohomology of abelian differentials, find new stratifications.
problem Cohomology classes in abelian differentials strata.
method Unified topological approach to find explicit stratifications.
result Explicit stratification of spin moduli space for odd spin structures.
Study on stable maps from 3-sphere to 3-space, focusing on singularities and invariants.
problem Understanding the behavior of stable maps from S3 to R3. method Analysis of codimension-one transitions, singular set behavior, and global invariants.
result Effects of decompositions on global invariants with prescribed branch sets.
Study irregular behavior of ball averages for non-amenable group actions on foliations.
problem Exploring irregular behavior of ball averages for non-amenable group actions.
method Introducing a new mechanism based on group structure to analyze irregular behavior.
result First examples of codimension one foliations with non-existent length averages.
We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyper…
Sharp pinching conditions restrict the geometry and topology of submanifolds.
problem Understanding submanifolds under pinching conditions in arbitrary Riemannian manifolds.
method Analyzing submanifolds with pinching conditions involving second fundamental form and mean curvature.
result The pinching condition imposes strong geometric and topological restrictions on submanifolds.
We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among…
We produce examples of codimension one foliations of the Euclidean and hyperbolic planes with bounded geometry which are topologically products, but for which leaves are non-recursively distorted. That is, the function which compares intrinsic distances in leaves with extrinsic distances in the ambient space grows fast…
To an inclusion topological groups H->G, we associate a naive G-spectrum. The special case when H=G gives the dualizing spectrum D_G introduced by the author in the first paper of this series. The main application will be to give a purely homotopy theoretic construction of Poincare embeddings in stable codimension.
We summarize our geometric and topological description of compact eight-manifolds which arise as internal spaces in N=1 flux compactifications of M-theory down to AdS3, under the assumption that the internal part of the supersymmetry generator is everywhere non-chiral. Specifying such a supersymmet…
Study defines a new invariant for foliated manifolds, linking index of Dirac operator to cohomology.
problem Defining a new invariant for foliated manifolds.
method Using transversal Seiberg-Witten map and finite dimensional approximation.
result Estimates and vanishing conditions for the index of transverse Dirac operator.