arXiv research

We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…

Constructs symplectic surface bundles with positive signatures.

We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the ma…

Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.

Study finds infinitely many surface bundle types with zero signature.

New examples of surface bundles found over surfaces.

Proves stability of certain vector bundles on Kähler surfaces.

Turaev-Viro invariants match for certain surface bundles.

Characterizes special curves on surface tangent bundles.

We show that there exist flat surface bundles with closed leaves having non-trivial normal bundles. This leads us to compute the Abelianisation of surface diffeomorphism groups with marked points. We also extend a formula of Tsuboi that expresses the Euler class of a flat circle bundle in terms of the Calabi invariant …

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on D…

Study proves correspondence for special bundles on complex surfaces.

Study Euler class of surface bundles with nontrivial results.

Hermitian bundle gerbes with connection are geometric objects for which a notion of surface holonomy can be defined for closed oriented surfaces. We systematically introduce bundle gerbes by closing the pre-stack of trivial bundle gerbes under descent. Inspired by structures arising in a representation theoretic approa…

We study the connected components of the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin …

For non-degenerate surfaces in $R^4$, a distinguished transversal bundle called affine normal plane bundle was proposed in [Nomizu-Vrancken]. Lagrangian surfaces have remarkable properties with respect to this normal bundle, like for example, the normal bundle being Lagrangian. In this paper we characterize those surfa…

Usually bundle gerbes are considered as objects of a 2-groupoid, whose 1-morphisms, called stable isomorphisms, are all invertible. I introduce new 1-morphisms which include stable isomorphisms, trivializations and bundle gerbe modules. They fit into the structure of a 2-category of bundle gerbes, and lead to natural d…

In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each $n \ge 3$ we construct $4$-manifolds $E$ admitting at least $n$ distinct fiberings $p_i: E \to Σ_{g_i}$ as a surface bundle over a surface with base and fiber both closed surfaces of negative Eule…

Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.

Surface bundles' signatures tied to their Euler characteristics.

We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in $S^n$ must locate in some $S^3\subset S^n$, from which we characterize Clifford torus as the only non-equatorial homogeneous minimal surface in $S^n$ with flat normal…

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minim…

For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $Σ$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in $M$. $S$ is embedded and is a section of the restriction of the bundle to the compleme…

Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.

The study identifies surfaces with Maslovian normal bundles.

Criterion for Lie algebroid connections on compact Riemann surfaces.

It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with ge…

Introduces Fock bundles for studying surface group character varieties.

Every surface bundle with genus $g$ fiber has a canonical Heegaard splitting of genus $2g+1$. We classify the mapping class groups of such Heegaard splittings in the case when the surface bundle has a sufficiently complicated monodromy map.

In arXiv:math/0605587, the first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface. This space can be identified with the real locus of the space of connections on the pullback of this bundle over …

Develops theory of d-holomorphic connections on Klein surfaces.

Investigate pseudoconvexity of locally trivial holomorphic ball bundles over compact Riemann surfaces.

The paper derives new Gauss-Bonnet formulas for frontal bundles over surfaces with boundary.

New method for decomposing manifolds into 1-handlebodies.

In the present note we describe geometrically the homology classes in the total space of a surface bundle over a surface in terms of the holonomy map. We treat the cases where the base surface is closed or has one boundary component. We replace the wrong theorem of the previous version by some observations on the homol…

The paper studies special surfaces in 4D space forms with specific geometric properties.

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the o…

An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…

Classifies $T^2$-bundles over closed surfaces up to isomorphisms.

We describe the second integral cohomology group of a surface bundle as the group of Chern classes of fiberwise holomorphic complex line bundles and use this to obtain information on this group.

Unified study of surfaces using Clifford algebras.

Researchers calculate Hofer-Zehnder capacity for twisted tangent bundles over surfaces.

We classify incompressible, boundary-incompressible, nonorientable surfaces in punctured-torus bundles over $S^1$. We use the ideas of Floyd, Hatcher, and Thurston. The main tool is to put our surface in the "Morse position" with respect to the projection of the bundle into the basis S^1.

Pseudo-Anosov subgroups in surface bundles over tori are convex cocompact.

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show…

Solves Dirac equation coupled to vector bundles.

The signature of a surface bundle over a surface is known to be divisible by 4. It is also known that the signature vanishes if the fiber genus is less than or equal to 2 or the base genus is less than or equal to 1. In this article, we construct new smooth 4-manifolds with signature 4 which are surface bundles over su…

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.