arXiv research

Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive hom…

The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1)_K, obtained by knot surgery on the rational elliptic surface E(1) with the left-handed trefoil knot K. E(1)_K has the simplest homotopy type among simply-connect…

Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.

Homotopy classification for certain 4-manifolds with dihedral fundamental groups.

The article proves estimates on homotopy and cohomology dimensions in fibrations.

Study shows weak homotopy equivalences for complete minimal surfaces.

The h-principle helps solve complex geometric problems.

Study shows conditions for rational ellipticity of manifolds with symmetries.

Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov's higher signatures on closed manifolds; - the problem of cut-and-paste invariance of Novikov's higher signatures on closed manifolds; - the problem of defining higher signa…

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…

We characterise simply-connected biquotients which potentially admit metrics of holonomy G_2. We prove that there are at most three real homotopy types of rationally elliptic such manifolds---all of them being formal. In the course of this examination we classify rationally elliptic homotopy types and characterise 7-di…

We investigate a parabolic-elliptic system for maps $(u,v)$ from a compact Riemann surface $M$ into a Lorentzian manifold $N\times{\mathbb{R}}$ with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the $v$-equation as a nonlinear second order constraint along…

In this short article we give a criterion whether a given minimal symplectic 4-manifold with $b_{2}^{+}=1$ having a torsion-free canonical class is rational or ruled. As a corollary, we confirm that most of homotopy elliptic surfaces $E(1}_{K}$, K is a fibered knot in $S^3$, constructed by R. Fintushel and R. Stern are…

We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give par…

In this paper we prove a tertiary index theorem which relates a spectral geometric and a homotopy theoretic invariant of an almost complex manifold with framed boundary. It is derived from the index theoretic and homotopy theoretic versions of a complex elliptic genus and interestingly related with the structure of the…

The paper proves properties of manifolds under certain conditions.

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…

Study finds negatively curved spheres in elliptic surfaces and their modifications.

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.

Cyclification of orbifolds explained in cohesive higher topos theory.

Characterizes totally elliptic surface group representations into Lie groups.

Elliptic surfaces have unique Lefschetz pencils and Calabi-Yau diffeomorphisms.

Elliptic surfaces without 1-handles proven for specific cases.

Study shows knots surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.

Study of 17 surface behaviors and singularities for elliptic Weingarten equations.

Characterizes compact complex surfaces with finite homotopy rank-sum.

Simple constructions of semi-discrete and discrete surfaces using Jacobi elliptic functions.

Estimates heights of special surfaces in warped products.

Study homotopy motions of surfaces in 3-manifolds.

Characterizes Stein surfaces with finite homotopy rank-sum.

Study homological mirror symmetry for Hirzebruch surfaces using Morse homotopy.

The paper studies elliptic operators on manifolds with boundary.

In the present paper we consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respec…

Study of area minimizing surfaces in homotopy classes of maps.

Defines homotopy type for links in thickened surfaces.

A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…

Constructs a new type of metric for elliptic surfaces.

We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fe…

The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

Researchers classify cmc surfaces using Jacobi elliptic functions.

Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipt…

In this paper we study the general affine differential geometry of surfaces in affine space $A^3$. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to aff…

Harer-Kas-Kirby conjectured that every handle decomposition of the elliptic surface E(1)_{2,3} requires both 1- and 3-handles. We prove that the elliptic surface E(n)_{p,q} has a handle decomposition without 1-handles for $n\geq 1$ and (p,q)=(2,3),(2,5),(3,4),(4,5).

This paper constructs explicit trisection diagrams for elliptic surfaces.

Paper calculates homotopy types of non-compact surfaces using groupoids.

Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.