arXiv research

We prove a Freed-Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa--Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the $PU(2)$-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms…

Characterizes smooth functions on manifolds with simple Reeb spaces.

We prove that for each closed smooth spin 4-manifold M there exists a closed smooth 4-manifold N such that the connected sum M # N admits a conformally flat Riemannian metric.

Study on 4-manifolds with exotic smooth structures and Z_2 fundamental group.

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the clo…

The $(α,β)$-Ricci-Yamabe flow exists on closed manifolds.

Multisections generalize Heegaard splittings and trisections to higher dimensions.

The paper explores the structure of Reeb spaces for smooth functions on manifolds.

The paper smooths out equations on 4-manifolds to show smooth moduli spaces.

In this paper, we prove a geometrization conjecture, every orientable smooth closed 3-manifold with finite fundamental group is homeomorphic to $S^3/G$ for some finite cyclic subgroup $G\subset {Isom}^+(S^3)$.

Smooth actions of cyclic groups on 3-manifolds are conjugate to smooth ones.

In this paper it is hown that given any smooth, positive function f on a closed, smooth manifold of dimension greater than four and with positive Paneitz invariant, there exists a metric on M such that $Q_g$ = f.

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed $n$-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifold…

Embeds 3-manifolds in symplectic 4-manifolds with constraints.

We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds. As a consequence, there are finitel…

Every 4-manifold can be smoothly embedded in complex projective 3-space.

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed conn…

We show that on a closed smooth manifold $M$ equipped with $k$ fiber bundle structures whose vertical distributions span the tangent bundle, every smooth diffeomorphism $f$ of $M$ sufficiently close to the identity can be written as a product $f=f_1... f_k$, where $f_i$ preserves the $i^{\text{th}}$-fiber. The factors …

For a smooth function on a smooth manifold of a suitable class, the space of all connected components of preimages is the graph and called the {\it Reeb graph}. Reeb graphs are fundamental tools in the algebraic and differential topological theory of Morse functions and more general functions which are not so wild. In …

Local minimizers are convex and close to Wulff shapes.

A flow defined by a nonsingular smooth vector field $X$ on a closed manifold $M$ is said to be parameter rigid if given any real valued smooth function $f$ on $M$, there are a smooth funcion $g$ and a constant $c$ such that $f=X(g)+c$ holds. We show that the parameter rigid flows on closed orientable 3-manifolds are sm…

The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.

Paper introduces fat CW complexes including all closed manifolds.

We show that every smooth closed oriented four-manifold admits a decomposition into two co- dimension zero submanifolds with common boundary. Each of these submanifolds carries a structure of a symplectic manifold with pseudo-convex boundary. This imply, in particular, that every smooth closed simply-connected four-man…

The minimal number of critical points is studied for smooth functions on closed manifolds.

Smooth manifolds have equivalent diffeomorphism groups if and only if they are diffeomorphic.

Multisections generalize trisections for 4-manifolds, allowing complex operations and explicit diagrams.

We classify smooth locally free actions of the real affine group on closed orientable three-dimensional manifolds up to smooth conjugacy. As a corollary, there exists a non-homogeneous action when the manifold is the unit tangent bundle of a closed surface with a hyperbolic metric.

Closed 4-manifolds foliated by hyperplanes are homeomorphic to the 4-torus.

In an earlier paper we showed that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. In that paper, we also introduced a new class of Lagrangian-type 4-dimensional su…

The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.

In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,α}$-close to the given sub manifold. We show also a version …

The hyperbolization process affects the structure of manifolds.

A gap in the proof of the main result in reference [1] in our original submission propagated into the constructions presented in the first version of our manuscript. In this version we give an alternative proof for the existence of Riemannian metrics with positive Ricci curvature on an infinite subfamily of closed, sim…

Classifies actions of SL(n,R) and SL(n,Z) on closed n-manifolds.

The Morse complex is shown to be an infinite functor.

We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic…

New 4D shapes found that defy smoothness rules.

Smooth 4-manifolds have simple horizontal decompositions.

The paper constructs infinitely many $G$-smoothings of a $G$-manifold.

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices…

By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan manifold without boundary. Applications to Sobolev spaces, Nash embedding and loca…

We prove that there are infinitely many pairs of homeomorphic non-diffeomorphic smooth 4-manifolds, such that in each pair one manifold admits an Einstein metric and the other does not. We also show that there are closed 4-manifolds with two smooth structures which admit Einstein metrics with opposite signs of the scal…

We study the classification of closed, smooth, spin, $1$-connected $7$-manifolds whose integral cohomology ring is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$. We also prove that if the integral cohomology ring of a closed, smooth, spin, $1$-connected $7$-manifold is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$ or $H^…

Notes on embedding criteria for smooth manifolds.

The paper classifies smooth structures on product manifolds of 3-connected 8-manifolds with spheres.

New method constructs smooth functions with specific Reeb graphs and preimages on 3D manifolds.

The study explores smooth structures on specific four-manifolds with cyclic groups, finding many admit infinitely many smooth structures.