Research

Research

Research Interests:

Differential Geometry, Geometric Analysis and Mathematical Physics.

My research is related to the geometry and topology of manifolds with special holonomy and calibrated geometry, analysis of deformation and elliptic theory on noncompact manifolds, high dimensional complex geometry and singularity theory, supersymmetry and Calabi-Yau manifolds, special Lagrangian submanifolds and the applications in mirror symmetry, associative and coassociative submanifolds of G_2 manifolds and the connections with M-Theory.

Publications:

Contact Structures on G_2-Manifolds and Spin 7-Manifolds

(with M. Firat Arikan and Hyunjoo Cho) (In preparation) (available at http://xxx.lanl.gov/abs/1207.2046)

Abstract: We show that there exist infinitely many pairwise distinct non-closed G_2-manifolds (some of which have holonomy full G_2) such that they admit co-oriented contact structures and have co-oriented contact submanifolds which are also associative. Along the way, we prove that there exists a tubular neighborhood N of every orientable three-submanifold Y of an orientable seven-manifold with spin structure such that for every co-oriented contact structure on Y, N admits a co-oriented contact structure such that Y is a contact submanifold of N. Moreover, we construct infinitely many pairwise distinct non-closed seven-manifolds with spin structures which admit co-oriented contact structures and retract onto co-oriented contact submanifolds of co-dimension four.

Remarks on Hamiltonian Structures in G_2-Geometry

(with Hyunjoo Cho and Albert J. Todd) (http://xxx.lanl.gov/abs/1309.1984, to appear in Journal of Mathematical Physics)

Abstract: In this article, we treat G_2-geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G_2-structure; in particular, we discuss existence and make a number of identifications of the spaces of Hamiltonian structures associated to the two multisymplectic structures associated to an integrable G_2-structure. Along the way, we prove some results in multisymplectic geometry that are generalizations of results from symplectic geometry.

Diffeomorphisms of 7-Manifolds with Coclosed G_2-Structure

(with Albert J. Todd) (http://xxx.lanl.gov/abs/1212.2261. (Part 2 of the sequel, to appear in Jour. of Math. Physics)

Abstract: We introduce coG_2-vector fields, coRochesterian 2-forms and coRochesterian vector fields on manifolds with a coclosed G_2-structure as a continuous of work from [15], and we show that the spaces of coG_2-vector fields and of coRochesterian vector fields are Lie subalgebras of the Lie algebra of vector fields with the standard Lie bracket. We also define a bracket operation on the space of coRochesterian 2-forms associated to the space of coRochesterian vector fields and prove, despite the lack of a Jacobi identity, a relationship between this bracket and so-called coG_2-morphisms.

Existence of Compatible Contact Structures on $G_2$-manifolds

(with M. Firat Arikan and Hyunjoo Cho) (Asian Journal of Math., Volume 17, No. 2, (2013), pp. 321-334.)

Abstract: In this paper, we show the existence of (co-oriented) contact structures on certain classes of G_2-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with G_2-structure) admits an almost contact structure. We also construct explicit almost contact metric structures on manifolds with G_2-structures.

Diffeomorphisms of 7-Manifolds with Closed G_2-Structure

(with Hyunjoo Cho and Albert J. Todd) (http://arxiv.org/abs/1112.0832, Part 1 of the sequel, to appear in Jour. of Math. Physics)

Abstract: We introduce G_2-vector fields, Rochesterian 1-forms and Rochesterian vector fields on manifolds with a closed G_2-structure as analogues of symplectic vector fields, Hamiltonian functions and Hamiltonian vector fields respectively, and we show that the spaces of G_2-vector fields and of Rochesterian vector fields are Lie subalgebras of the Lie algebra of vector fields with the standard Lie bracket. We also define, in analogy with the Poisson bracket on smooth real-valued functions from symplectic geometry, a bracket operation on the space of Rochesterian 1-forms associated to the space of Rochesterian vector fields and prove, despite the lack of a Jacobi identity, a relationship between this bracket and diffeomorphisms which preserve G_2-structures.

A Note on Closed G_2-Structures and 3-Manifolds

(with Hyunjoo Cho and Albert J. Todd) (http://xxx.lanl.gov/abs/1112.0830, to appear in Turkish Journal of Math)

Abstract: This article shows that given any orientable 3-manifold X, the 7-manifold T^*X x R admits a closed G_2-structure varphi=Re(Omega)+omega\wedge dt where Omega is a certain complex-valued 3-form on T^*X; next, given any 2-dimensional submanifold S of X, the conormal bundle N^*S of S is a 3-dimensional submanifold of T^*X x R such that varphi restricted to N^*S is equivalent to 0. A corollary of the proof of this result is that N^*S x R is a 4-dimensional submanifold of T^*X x R such that varphi restricted to N^*S x R is equivalent to 0.

Deformations of Asymptotically Cylindrical Special Lagrangian Submanifolds with Moving Boundary

(with Albert J. Todd) (Part 2, Proc. Gokova Geometry-Topology, (2010), 99-124, Int. Press.)

Abstract: In an earlier paper, we proved that, under certain hypotheses, the moduli space of an asymptotically cylindrical special Lagrangian submanifold with fixed boundary of an asymptotically cylindrical Calabi-Yau 3-fold is a smooth manifold. Here we prove the analogous result for an asymptotically cylindrical special Lagrangian submanifold with moving boundary.

Deformations of Asymptotically Cylindrical Special Lagrangian Submanifolds with Fixed Boundary

(with Albert J. Todd) (Part 1, Proc. Gokova Geometry-Topology, (2010), 99-124, Int. Press.)

Abstract: Given an asymptotically cylindrical special Lagrangian submanifold L in an asymptotically cylindrical Calabi-Yau 3-fold X, we determine conditions on a decay rate gamma which make the moduli space of (local) special Lagrangian deformations of L in X a smooth manifold and show that it has dimension equal to the dimension of the image of H^1_{cs}(L,R) in H^1(L,R) under the natural inclusion map.

Mirror Duality in a Joyce Manifold

(with Selman Akbulut and Baris Efe) (Advances in Mathematics, 223, (2010), pp. 444-453)

Abstract: Previously the two of the authors defined a notion of dual Calabi-Yau manifolds in a G_2 manifold, and described a process to obtain them. Here we apply this process to a compact G_2 manifold, constructed by Joyce, and as a result we obtain a pair of Borcea-Voisin Calabi-Yau manifolds, which are known to be mirror duals of each other.

New Spin(7) holonomy metrics admiting G_2 holonomy reductions and M-theory/IIA dualities.

(with Osvaldo Santillan) (Phys. Rev. D., 79, (2009), no. 8, 086009)

Abstract: As is well known, when D6 branes wrap a special lagrangian cycle on a non compact CY 3-fold in such a way that the internal string frame metric is Kahler there exists a dual description, which is given in terms of a purely geometrical eleven dimensional background with an internal metric of G_2 holonomy. It is also known that when D6 branes wrap a coassociative cycle of a non compact G_2 manifold in presence of a self-dual two form strength the internal part of the string frame metric is conformal to the G_2 metric and there exists a dual description, which is expressed in terms of a purely geometrical eleven dimensional background with an internal non compact metric of Spin(7) holonomy. In the present work it is shown that any G_2 metric participating in the first of these dualities necessarily participates in one of the second type. Additionally, several explicit Spin(7) holonomy metrics admitting a G_2 holonomy reduction along one isometry are constructed. These metrics can be described as R-fibrations over a 6-dimensional Kahler metric, thus realizing the pattern Spin(7) to G_2 to (Kahler) mentioned above. Several of these examples are further described as fibrations over the Eguchi-Hanson gravitational instanton and, to the best of our knowledge, have not been previously considered in the literature.

Calibrated associative and Cayley embeddings.

(with Colleen Robles) (Asian Journal of Math. Volume 13, No. 3, (2009), pp. 287-306)

Abstract: Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold. Second, the interior of a smooth, compact 4-manifold K, whose double has a trivial bundle of self-dual 2-forms, may be isometrically embedded into a Spin(7)-manifold as a Cayley submanifold. Along the way, we also show that Bochner's Theorem on real analytic approximation of smooth differential forms, can be obtained using real algebraic tools developed by Akbulut and King.

Mirror Duality via G_2 and Spin(7) Manifolds

(with Selman Akbulut) (Arithmetic and Geometry Around Quantization, Progress in Math., Birkhauser, 2009-2010) (also available at math archives)

Abstract: The main purpose of this paper is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,\phi,\Lambda), with the calibration 3-form \phi and an oriented 2-plane field \Lambda, a pair of parametrized tangent bundle valued 2 and 3-forms of M. These forms can then be used to define various different complex and symplectic structures on certain 6-dimensional subbundles of T(M). When these bundles are integrated they give mirror CY manifolds. In a similar way, one can define mirror dual G_2 manifolds inside of a Spin(7) manifold (N^8, \Psi). In case N^8 admits an oriented 3-plane field, by iterating this process we obtain Calabi-Yau submanifold pairs in N whose complex and symplectic structures determine each other via the calibration form of the ambient G_2 (or Spin(7)) manifold.

Mirror symmetry aspects for compact G_2 manifolds

(with Osvaldo Santillan) (preprint) (also available at math archives)

Abstract: The present paper deals with mirror symmetry aspects of compact ``barely'' G_2 manifolds, that is, G_2 manifolds of the form (CY x S^1)/Z_2. We propose that the mirror of any barely G_2 manifold is another barely one and which is constructed as a fibration of the ``mirror’’ of the CY base. Also, we describe the Joyce manifolds of the first kind as ``barely'' and we show that the underlying CY of all the family is self-mirror with h^{1,1}=h^{2,1}=19. We thus propose that the mirror of a Joyce space of the first kind will be another Joyce space of the first kind.We also suggest that this self-mirror CY family is dual to K3x S^1 in the heterotic/M-theory sense, and that arise as a particular case of the Borcea-Voisin construction. As a spin-off we conclude from this analysis that no 5-brane instantons are present in compactifications of eleven dimensional supergravity over Joyce manifolds of the first kind.

Deformations in G_2 Manifolds.

(with Selman Akbulut) (Advances in Mathematics, Volume 217, Issue 5, 20, Pages 2130-2140, 2008)

Abstract: Here we study the deformations of associative submanifolds inside a G_2 manifold M^7 with a calibration 3-form \phi. A choice of 2-plane field \Lambda on M (which always exits) splits the tangent bundle of M as a direct sum of a 3-dimensional associate bundle and a complex 4-plane bundle TM= E\oplus V, and this helps us to relate the deformations to Seiberg-Witten type equations. Here all the surveyed results as well as the new ones about G_2 manifolds are proved by using only the cross product operation (equivalently \phi). We feel that mixing various different local identifications of the rich G_2 geometry (e.g. cross product, representation theory and the algebra of octonions) makes the study of G_2 manifolds looks harder then it is (e.g. the proof of McLean's theorem \cite{m}). We believe the approach here makes things easier and keeps the presentation elementary. This paper is essentially self contained.

Deformations of Asymptotically Cylindrical Coassociative Submanifolds with Moving Boundary.

(preprint)(also available at math archives)

Abstract: In a recent paper, we proved that given an asymptotically cylindrical G_2-manifold M with a Calabi-Yau boundary X, the moduli space of coassociative deformations of an asymptotically cylindrical coassociative 4-fold C in M with a fixed special Lagrangian boundary L in X is a smooth manifold of dimension dim (V_+), where V_+ is the positive subspace of the image of H^2_{cs}(C,R) in H^2(C,R). In order to prove this we used the powerful tools of Fredholm Theory for noncompact manifolds which was developed by Lockhart and McOwen and independently by Melrose.

In this paper, we extend this result to the moving boundary case. Let Phi:H^2(L,R)à H^3_{cs}(C,R) be the natural projection, so that ker(Phi) is a vector subspace of H^2(L,R). Let F be a small open neighbourhood of 0 in ker(Phi). Here we prove that the moduli space of coassociative deformations of an asymptotically cylindrical coassociative submanifold C asymptotic to L_s x (R,infty), for s in F, is a smooth manifold of dimension equal to dim (V_+)+dim(ker(Phi))=dim (V_+)+b^2(L)-b^0(L)+b^3(C)-b^1(C)+b^0(C).

Calibrated Manifolds and Gauge Theory.

(with Selman Akbulut) (J. Reine Angew. Math., Crelle's J., Volume 2008, Issue 625, Pages 187–214, December 2008)

Abstract: The purpose of this paper is to relate the geometries of calibrated submanifolds to their gauge theories. We study the moduli space of deformations of a special kind of associative submanifolds in a G_2 manifold (which we call complex associative submanifolds); and we study the moduli space of deformations of a special kind of Cayley submanifolds (which we call complex Cayley submanifolds). We show that deformation spaces can be perturbed to be smooth and finite dimensional. We also get similar results for the deformation spaces of other calibrated submanifolds. We discuss the relation to Seiberg-Witten theory, and propose a certain counting invariant for associative and Cayley submanifolds of foliated manifolds.

Deformations of Special Lagrangian Submanifolds; An Approach via Fredholm Alternative.

(Gokova Geometry-Topology (2005), No 1, 154-161, International Press, 2006)

Abstract: In an earlier paper, we showed that the moduli space of deformations of a smooth, compact, orientable special Lagrangian submanifold L in a symplectic manifold X with a non-integrable almost complex structure is a smooth manifold of dimension H^1(L), the space of harmonic 1-forms on L. We proved this first by showing that the linearized operator for the deformation map is surjective and then applying the Banach space implicit function theorem. In this paper, we obtain the same surjectivity result by using a different method, the Fredholm Alternative, which is a powerful tool for compact operators in linear functional analysis.

Deformations of Asymptotically Cylindrical Coassociative Submanifolds with Fixed Boundary.

(with Dominic Joyce) (Geometry&Topology, Vol. 9 (2005) Paper no. 25, 1115—1146)

Abstract: McLean proved that the moduli space of coassociative deformations of a compact coassociative 4-submanifold C in a G_2-manifold M is a smooth manifold of dimension equal to b^2_+(C). In this paper, we show that the moduli space of coassociative deformations of a noncompact, asymptotically cylindrical coassociative 4-fold C in an asymptotically cylindrical G_2-manifold M is also a smooth manifold. Its dimension is the dimension of the positive subspace of the image of H^2_{cs}(C,R) in H^2(C,R).

Associative submanifolds of a G_2 manifold.

(with Selman Akbulut) (this paper is part of Advances in Mathematics, Volume 217, Issue 5, Pages 2130-2140, 2008)(also available at math archives)

Abstract: We study deformations of associative submanifolds of a G_2 manifold. We show that deformation spaces can be perturbed to be smooth and finite dimensional, and they can be made compact by constraining them with an additional equation and reducing it to Seiberg-Witten theory. This allows us to associate invariants of certain associative submanifolds. More generally we apply this process to certain associated Grassmann bundles in order to assign invariants to G_2 manifolds.

Asymptotically Cylindrical Ricci-Flat Manifolds.

(Proceedings of American Mathematical Society, 134 (2006), no. 10, 3049-3056)

Abstract: Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindrical manifold (M,g) with Ricci-flat metric g can have at most two cylindrical ends. In the case where there are two such cylindrical ends then there is reduction in the holonomy group Hol(g) and (M,g) is a cylinder.

Deformations of Special Lagrangian Submanifolds.

(Communications in Contemporary Mathematics 2 (2000), no. 3, 365--372)

Abstract: R.C.McLean showed that the moduli space of nearby submanifolds of a smooth, compact, orientable special Lagrangian submanifold L in a Calabi-Yau manifold X is a smooth manifold and its tangent space at L is identified with the space of harmonic one forms on L. In this paper, we extend this result from Calabi-Yau manifolds to the symplectic manifolds with non-integrable almost complex structure.