Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.
problem Understanding spaces of positive scalar curvature metrics on totally nonspin manifolds.
method Analysis of positive scalar curvature metrics on manifolds with spin boundary, using propagation techniques.
result Spaces of positive scalar curvature metrics are not connected and have nontrivial fundamental groups for certain dimensions.
The geography and botany of smooth/symplectic nonspin 4-manifolds with abelian fundamental group are addressed.
Motivated by Witten's spinor proof of the positive mass theorem, we analyze asymptotically constant harmonic spinors on complete asymptotically flat nonspin manifolds with nonnegative scalar curvature.
Nonspin manifolds can't have uniformly positive scalar curvature.
problem Proving nonexistence of uniformly positive scalar curvature metrics on nonspin manifolds.
method Using Schoen-Yau's minimal hypersurface results and properties of maps to product spaces.
result Enlargeable metrics on nonspin manifolds cannot have uniformly positive scalar curvature.
In \cite{AP3, AHP}, the first author and his collaborators constructed the irreducible symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n−1)CP2#(2n−1)CP2 for each integer n≥25, and the families of simply connected irreducible nonspin symplectic 4…
Let N be a closed enlargeable manifold in the sense of Gromov-Lawson and M a closed spin manifold of equal dimension, a famous theorem of Gromov-Lawson states that the connected sum M#N admits no metric of positive scalar curvature. We present a potential generalization of this result to the case where M is n…
Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain homological condition is satisfied. We present a counterexample to this conjecture, based…
Constructs metrics with positive scalar curvature on odd-order abelian fundamental group manifolds.
problem Creating metrics with positive scalar curvature on manifolds with odd-order abelian fundamental groups.
method Introduced Riemannian metrics, proved homology invariance, constructed metrics using admissible products.
result Solved the Gromov-Lawson-Rosenberg conjecture for manifolds with odd-order abelian fundamental groups.
New 5-manifold found with zero Ricci curvature.
problem Finding almost Ricci-flat manifolds with specific properties.
method Kummer-type constructions using Riemannian metrics.
result Constructed a simply connected, nonspin 5-manifold with zero Ricci curvature.
New symplectic 4-manifolds with non-negative signatures are constructed using complex surfaces and quotients.
problem Creating new symplectic 4-manifolds with non-negative signatures.
method Using complex surfaces, Cartwright-Steger surfaces, and Hirzebruch's line-arrangement surfaces, along with quotients.
result Irreducible symplectic and non-symplectic 4-manifolds homeomorphic but not diffeomorphic to (2n−1)CP2#(2n−1)CPˉ2 are constructed. 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.
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 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.
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.
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.
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.
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.
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
The paper explores F-manifolds and metrics, constructing canonical structures.
problem Understanding relationships between F-manifolds and metrics.
method Construction of canonical flat F-manifolds and homogeneous Riemannian F-manifolds.
result Construction of a canonical flat F-manifold associated to an arbitrary Riemannian F-manifold.
Defines s-manifolds and s-manifolds with corners for symplectic applications.
problem Creating a framework for singular manifolds with useful properties.
method Introducing categories of stratified manifolds and manifolds with corners.
result Fundamental classes and transverse fibre products in s-manifolds.
Extends holomorphic Cartan geometry to Sasakian manifolds.
problem No specific problem stated; extends geometry to new context.
method Extends holomorphic Cartan geometry to Sasakian manifolds.
result Investigated branched holomorphic Cartan geometries on Sasakian Calabi-Yau manifolds.
Embeds 3-manifolds in symplectic 4-manifolds with constraints.
problem Embedding 3-manifolds in symplectic 4-manifolds with topological and smooth properties.
method Topological and smooth embeddings, using homology cobordism and obstructions.
result 3-manifolds can be embedded in symplectic 4-manifolds with specific conditions.
A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold M~ with the deck transform group acting conformally on M~. If M admits a holomorphic flow, acting on M~ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vais…
The study provides homological characterizations for Q-manifolds and l2-manifolds.
problem Density of maps in characterizing Q-manifolds and l2-manifolds. method Investigates weakening the density of Zn-maps and Z-maps to homological maps. result Obtains homological characterizations for Q-manifolds and l2-manifolds. This paper provides a new method to construct b-symplectic toric manifolds from toric manifolds.
problem Classifying and constructing b-symplectic toric manifolds. method A new method to construct b-symplectic toric manifolds from toric manifolds. result This new method allows for the decomposition of b-symplectic toric manifolds into toric manifolds. We introduce a new general class of metric f-manifolds which we call (nearly) trans-S-manifolds and includes S- manifolds, C-manifolds, s-th Sasakian manifolds and generalized Kenmotsu manifold studied previously. We prove their main properties and we present many examples which justify their study.
Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.
problem Generalize K-contact and Sasakian manifolds criteria using weak quasi contact metric manifolds.
method Study weak quasi contact metric manifolds and generalize theorems for K-contact and Sasakian manifolds.
result Provide new criterions for K-contact and Sasakian manifolds in terms of curvature tensor and geometric objects.
The study classifies Kähler-Frobenius manifolds and their properties.
problem Classifying Kähler-Frobenius manifolds and understanding their structure.
method Using Topological Quantum Field Theory and Frobenius manifold theory.
result All flat compact Kähler manifolds are Frobenius manifolds and are classified.
Products of LCK manifolds do not admit LCK structures.
problem Whether products of compact complex manifolds admit LCK metrics.
method Classifying known LCK manifolds and proving non-existence of LCK structures in product cases.
result Products of LCK manifolds do not admit LCK structures.
Study geodesics on infinite-dimensional manifolds using Finsler structures.
problem Geodesics on Fréchet manifolds of Riemannian metrics.
method Establish Riemann-Finsler structures, prove existence and minimality of geodesics, derive Euler-Lagrange equations.
result Geodesics on Fréchet manifolds of Riemannian metrics are length minimizing and satisfy Euler-Lagrange equations.
New proof shows compact homogeneous LCK manifolds are Vaisman.
problem Proving compact homogeneous LCK manifolds are Vaisman.
method Using homogeneous LCK manifolds with potential and a new metric construction.
result Compact homogeneous LCK manifolds are Vaisman.
Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.
problem Existence of minimal surfaces in non-compact 3-manifolds.
method Analyzes specific cases of hyperbolic 3-manifolds with bounded geometry and rank-1 cusps.
result Proves existence of minimal surfaces in various hyperbolic 3-manifolds.
The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
problem Proving Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
method Analyzing maps between different classes of pseudo-Hermitian manifolds, using curvature assumptions.
result Holomorphic maps are constant under certain curvature conditions.
A universal branched 3-manifold W characterizes Sol 3-manifolds.
problem Characterizing Sol 3-manifolds
method Using a universal branched 3-manifold W and a regular language result A closed 3-manifold M immerses into W if and only if M admits a Sol structure. Study on hypersymplectic manifolds and their geometric properties.
problem Obstruction for hypersymplectic manifolds with SU(1,1) action.
method Analyzes geometric properties and actions of SU(1,1) on hypersymplectic manifolds.
result Hypersymplectic manifolds with vanishing obstruction are metric cones over split 3-Sasakian manifolds.
Proves almost flat spin^c manifolds bound compact manifolds.
problem Proving almost flat spin^c manifolds bound compact manifolds.
method Long-standing conjecture of Farrell--Zdravkovska and S. T. Yau settled.
result Every almost flat spin^c manifold bounds a compact orientable manifold.
Study Hodge theory and supersymmetry on Sasakian and Vaisman manifolds.
problem Decomposing harmonic forms and cohomology on Sasakian and Vaisman manifolds.
method Construct Lie superalgebra, use Hodge decomposition, compute supersymmetry algebra.
result Explicit computation of the supersymmetry algebra of Sasakian manifolds.
The paper characterizes graph manifolds using fold maps and embeddability of polyhedra.
problem Understanding the global topologies of graph manifolds.
method Using fold maps into the plane and embeddability of polyhedra in 3-manifolds.
result Characterizes graph manifolds via fold maps and polyhedra embeddability.
The paper defines quasi-isometry for almost contact metric manifolds and explores its properties.
problem Understanding quasi-isometry in almost contact metric manifolds.
method Definition and study of quasi-isometry for almost contact metric manifolds.
result Established a relation between scalar curvature and quasi-isometric constants.