arXiv subjects
Publications and source records attributed to Mustafa Kalafat.
In this paper, we compute the index and nullity for minimal submanifolds of some complex Einstein spaces. We investigate the stability of these minimal submanifolds and suggest a criterion for instability for some cases. We also compute some higher eigenvalues for the Laplacian of the Berger spheres and provide with an algorithm.
In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of manifolds with calibrated geometries. We collect these results here for the sake of completeness. As applications of our algebraic topological study we present some results on special Lagrangian-free embeddings of surfaces and 3-manifolds into the Euclidean 4 and 6-space.
We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal bundle. We also give another proof of the fact that a compact complex surface together with an Einstein-Hermitian metric of positive orthogonal bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini-Study metric up to rescaling. This result relaxes the Kähler condition in Berger's theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.
We prove a nonexistence theorem for product type manifolds. In particular we show that the 4-manifold $Σ_g\timesΣ_h$ does not admit any locally conformally flat metric arising from discrete and faithful representations for $g\geq 2$ and $h\geq 1$
We show that if a compact complex surface admits a locally conformally flat metric, then it cannot contain a smooth rational curve of odd self-intersection. In particular, the surface has to be minimal. Then we give a list of possibilities of such surfaces.
We exhibit a family of complex manifolds, which has a member at each odd complex dimension and which has the same cohomology groups as the complex projective space at that dimension, but not homotopy equivalent to it. We also analyze the even dimensional analogue.
We show that if a compact, oriented 4-manifold admits a coassociative-free immersion into the Euclidean 7-space then its Euler characteristic and signature vanish. Moreover, in the spin case the Gauss map is contractible, so that the immersed manifold is parallelizable. The proof makes use of homotopy theory in particular obstruction theory. As a further application we prove a non-existence result for some infinite families of 4-manifolds that can not be addressed previously. We give concrete examples of parallelizable 4-manifolds with complicated non-simply-connected topology.
In this paper, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces. Finally we give information on how the Page space is related to some other metrics on the same underlying smooth manifold.
In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G_2 manifolds with special properties, which appear in the first named author's work with S. Salur about G_2 dualities.
We show that a compact complex surface which admits a conformally Kähler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g becomes a multiple of the Fubini-Study metric.
This is a survey article on the existence of locally conformally flat(LCF) and self-dual(SD) metrics on various basic 4-manifolds like simply-connected ones or product types
We construct infinite families of non-simply connected locally conformally flat (LCF) 4-manifolds realizing rich topological types. These manifolds have strictly negative scalar curvature and the underlying topological 4-manifolds do not admit any Einstein metrics. Such 4-manifolds are of particular interest as examples of Bach-flat but non-Einstein spaces in the non-simply connected case. Besides that the underlying smooth manifolds are examples of spaces that admit open book decomposition in dimension 4.
We construct self-dual(SD) but not locally conformally flat(LCF) metrics on families of non-simply connected 4-manifolds with small signature. We construct various sequences with bounded or unbounded Betti numbers and Euler characteristic. These metrics have negative scalar curvature. As an application, this addresses the Remark 4.79 of Besse.
We analyze the symplectic and complex structures on the panelled web 4-manifolds. In particular, we give infinite family of examples of almost complex but not symplectic and not complex 4-manifolds in the non-simply connected case.
In this article, we prove that a quotient of a K3 surface by a free Z_2+Z_2 action does not admit any metric of positive scalar curvature. This shows that the scalar flat anti self-dual metrics (SF-ASD) on this manifold can not be obtained from a family of metrics for which the scalar curvature changes sign, contrary to the previously known constructions of this kind of metrics on manifolds of b+=0.
In this article, we give a survey of Geometric Invariant Theory for Toric Varieties, and present an application to the Einstein-Weyl Geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted projective space CP_(1,1,2). We also find and classify all possible quotients.
We show that a complete simply-connected hyperkaehler 4-manifold with an isometric triholomorphic circle action is obtained from the Gibbons-Hawking ansatz with some suitable harmonic function.
Under a vanishing hypothesis, Donaldson and Friedman proved that the connected sum of two self-dual Riemannian 4-Manifolds is again self-dual. Here we prove that the same result can be extended over to the positive scalar curvature case.