Local criteria for non-embeddability of complex manifolds.
problem Smooth non-embeddability of Levi-flat manifolds.
method Analogue of Ueda theory on neighborhood structure of hypersurfaces.
result Local criteria for non-embeddability of Levi-flat manifolds.
New result on Levi-flat hypersurfaces' normal bundles without positive curvature.
problem Understanding Levi-flat hypersurfaces' normal bundles and their curvature properties.
method Analyzing the normal bundle of Levi-flat real hypersurfaces in complex manifolds.
result The normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature.
Let X be an orientable compact Levi-flat CR manifold and let L be a positive CR complex line bundle over X. We prove that certain microlocal conjugations of the associated Szegő kernel admits an asymptotic expansion with respect to high powers of L. As an application, we give a Szegő kernel proof of the Kodaira…
We prove the Lefchetz theorem for CR submanifolds in Hermitian symmetric spaces. As an application we prove the nonexistence of real analytic Levi flat submanifolds in such manifolds.
A Lipschitz hypersurface is a hypersurface which locally is the graph of a Lipschitz function. A Lipschitz (or C^1) hypersurface is said to be Levi-flat if it is locally foliated by complex manifolds of complex dimension (n-1). We shall prove that there exist no Lipschitz Levi-flat hypersurfaces in CP^n with n >= 3. Ou…
Study relationships between submanifolds and ambient Kahler 4-manifolds' fundamental groups.
problem Relationships between submanifolds and fundamental groups of Kahler 4-manifolds.
method Analyzes fundamental groups of embedded Levi-flat or pseudoconvex submanifolds in Kahler 4-manifolds.
result Fundamental group of M4 determined by the fundamental group of compact embedded Levi-flat or pseudoconvex submanifolds. We prove a relation between the ∂ˉM cohomology of a minimal orbit M of a real form G0 of a complex semisimple Lie group G in a flag manifold G/Q and the Dolbeault cohomology of the Matsuki dual open orbit X of the complexification K of a maximal compact subgroup K0 of G0, under the assum…
The paper constructs Levi flat structures using structure sheaves and differential complexes.
problem Global solvability and regularity of Levi flat structures.
method Employing formal integrability and differential complexes, the paper constructs a resolution for the structure sheaf.
result Global exactness and Sobolev regularity of the differential complex for Levi flat structures.
Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.
problem Computing Dolbeault cohomology for Levi-flat CR structures on compact Lie groups.
method Algebraic classification of left-invariant CR structures combined with Pittie's result on compact Lie groups.
result Generalization of Dolbeault cohomology computation to Levi-flat CR structures.
We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat…
We focus on the topology and dynamics of minimal sets and Levi-flats in surfaces of general type. Our method relies on the ergodic theory of Riemann surfaces laminations: we use harmonic measures and Lyapunov exponents. Our first result establishes that minimal sets have large Hausdorff dimension when a leaf is simply …
In this paper we prove the following theorem. Main Theorem. Let n >= 3 and m >= 3n/2 +7. Then there exists no C^m Levi-flat real hypersurface M in P_n. The condition that M is Levi-flat means that when M is locally defined by the vanishing of a C^m real-valued function f, at every point of M the restriction of d d-bar …
Local conditions on boundaries of C∞ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the hypersurface to be real analytic. This allows us to classify all real analytic generic bou…
New condition extends Diederich-Fornæss index for pseudoconvex domains.
problem Determine sufficient conditions for Diederich-Fornæss index to be close to 1.
method Derive sufficient condition on Levi-flat sets of the boundary.
result Diederich-Fornæss index is 1 if Levi-flat sets are transversal to holomorphic tangent vector fields.
A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is stu…
We address the problem of existence and uniqueness of a Levi-flat hypersurface M in Cn with prescribed compact boundary S for n≥3. The situation for n≥3 differs sharply from the well studied case n=2. We first establish necessary conditions on S at both complex and CR points, needed for the existence…
We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in …
Let Ω be a pseudoconvex domain with C2-smooth boundary in CPn. We prove that the ∂ˉ−NeumannoperatorNexistsfor(p,q)−formsonΩ.Furthermore,thereexistsat_0>0suchthattheoperatorsN,\bar\partial^*N,\bar\partial N$ and the Bergman projection are regular in the Sobolev …
To be prudent, the paper has been withdrawn by the authors, due an error (missing complex conjugate sign) in Equation (2.5). We are very grateful to Marco Brunella for pointed out the error.
Paper constructs solutions for a class of overdetermined systems.
problem Constructing solutions for a class of overdetermined systems.
method Resolution of the solution sheaf, sufficient condition for global exactness, gluing techniques, local solvability of the Treves complex.
result Obtained a sufficient condition for global exactness, leading to gluing techniques for local solutions.
The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter…
The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.
problem Boundary estimates for solutions to fully non-linear elliptic equations on Hermitian manifolds.
method Unified approach using quantitative boundary estimates, gradient estimates, and existence results.
result Established gradient estimates and unified approach to Dirichlet problem solutions.
Characterizes hypersurfaces in complex projective and hyperbolic planes.
problem Geometric characterization of hypersurfaces.
method Geometric characterization and partial classifications.
result Characterizes certain hypersurfaces of cohomogeneity one.
In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Le…
A primary goal in this paper is to study the question that asks when a real analytic submanifold M in Cn+1 bounds a real analytic (up to M) Levi-flat hypersurface M^ near p∈M such that M^ is foliated by a family of complex hypersurfaces moving along the normal direction of M at …
New class of metric f-manifolds introduced.
problem No specific problem stated; focuses on introducing new class.
method Definition and properties of new class of metric f-manifolds.
result Properties and examples of new class of metric f-manifolds.
Characterizes and describes selfsimilar Hessian manifolds with homothetic vector fields.
problem Understanding the structure and properties of selfsimilar Hessian manifolds.
method Characterization and description of selfsimilar manifolds with homothetic vector fields.
result Any selfsimilar Hessian manifold with a potential homothetic vector field is locally isomorphic to a product of radiant Hessian manifolds.
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
The paper defines trans-para-Sasakian manifolds and explores their geometric properties.
problem Exploring the geometry of trans-para-Sasakian manifolds.
method Definition and study of curvature properties.
result Conditions for η−Einstein and Einstein manifolds. In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with (2n+s)-dimensional s−contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…
Study on a new type of manifolds that generalize almost C-manifolds.
problem Understanding weak nearly C-manifolds and their properties.
method Analyzing conditions for local Riemannian product structures and characterizing specific dimensions.
result Conditions for a weak nearly C-manifold to become locally a Riemannian product and characterization of specific dimensions.
New class of complex manifolds defined, properties studied.
problem Understanding properties of complex manifolds.
method Introduced and studied wHHR manifolds, proved metric equivalence.
result Bergman and Kobayashi metrics are biLipschitz equivalent for wHHR Stein manifolds.
Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.
problem Understanding the equivalence between stabilized convex symplectic manifolds and flexible Weinstein manifolds.
method Analyzing the homotopy type and symplectic properties of the manifolds.
result Stabilized convex symplectic manifolds are symplectomorphic to flexible Weinstein manifolds.
New criteria for Sasakian manifolds and classification of nearly cosymplectic manifolds.
problem Characterizing Sasakian and cosymplectic manifolds.
method Analyzing nearly Sasakian and nearly cosymplectic manifolds of dimensions greater than five.
result Every nearly Sasakian manifold of dimension greater than five is Sasakian.
Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.
problem Embedding 3-manifolds smoothly in 5-manifolds.
method Homotopy and small homotopy to achieve smooth embeddings.
result Locally flat embeddings are homotopic to smooth ones.
Determines the type of 3-manifolds from their covers.
problem Identifying the type of 3-manifolds.
method Uses Thurston seminorms of covers.
result Finite covers determine the type of 3-manifolds.
The study identifies criteria for 3-manifolds to be boundaries of exotic 4-manifolds.
problem Determining which 3-manifolds can be boundaries of exotic 4-manifolds.
method Provided criteria and examples of 3-manifolds that can be boundaries of 4-manifolds with infinitely many distinct smooth structures.
result Identified specific types of 3-manifolds (weakly fillable contact, non-vanishing Heegaard Floer invariant) that are boundaries of 4-manifolds with infinitely many distinct smooth structures.
The study provides a structure theorem for a new class of noncompact 3-manifolds.
problem Understanding a new class of noncompact 3-manifolds.
method Proved a structure theorem for irreducible open graph manifolds.
result A canonical 'reduced' decomposition of irreducible open graph manifolds along embedded, incompressible 2-tori.
Compact Vaisman manifolds split into simpler parts.
problem Understanding structure of compact Vaisman manifolds.
method Extending splitting result to compact Vaisman manifolds.
result Compact Vaisman manifolds are finitely covered by a product of a Sasakian manifold and a circle.
Condition for intersection of real flag manifolds in complex flag manifold.
problem Intersection conditions of real flag manifolds in a complex flag manifold.
method Condition given in terms of symmetric triad, antipodal intersection proven.
result Intersection of real flag manifolds is antipodal.
The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.
problem Exploring new types of manifolds in general relativity.
method Generalization of existing manifolds and construction of specific examples.
result Existence and properties of extended quasi-Einstein manifolds with solitons.
The paper explores toric Vaisman manifolds and their connections to Sasaki and Kähler geometry.
problem Understanding the geometric relationships between Vaisman, Sasaki, and Kähler manifolds in the toric context.
method Introducing and analyzing toric Vaisman structures, showing relationships between minimal coverings and associated Sasaki manifolds, and proving conditions for toricity.
result Toric Vaisman manifolds have a close relationship with toric Sasaki manifolds, and vice versa, under specific conditions.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
problem Classifying 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
method Classifies vertical 3-manifolds as preimages of arcs on the plane for simplified (2,0)-trisection maps.
result Each 6-tuple of vertical 3-manifolds determines the source 4-manifold uniquely up to orientation reversing diffeomorphisms.
Study on 3D manifolds with specific tensor structures and their properties.
problem Characterizing 3D Riemannian manifolds with tensor structures.
method Investigation of locally conformal Riemannian product manifolds and their associated structures.
result Conditions for additional structures to be parallel and properties of almost Einstein and Einstein manifolds.
New manifold type PNDP-manifold defined with Einstein warped product structure.
problem Defining manifolds with non-standard dimensions.
method Einstein warped product manifold with special base and fiber structures.
result PNDP-manifolds are Einstein warped product manifolds with specific base and fiber properties.
Constructs higher-dimensional twistor spaces for complex manifolds.
problem Creating generalized twistor spaces for complex manifolds.
method Generalizing twistor spaces to more complex manifolds and constructing branched double covers.
result Non-Kahler Calabi-Yau manifolds can be constructed from these generalized spaces.
Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds
problem Establishing optimal inequalities relating systole, inradius, and volume in hyperbolic 3-manifolds
method Using systole-volume inequalities for extremal manifolds
result Extremal manifolds for systole, inradius, and volume are identified