arXiv research

Model for associative submanifolds in K3 fibrations.

This study addresses transitions in conically singular associative submanifolds and their desingularizations.

Study constructs associative submanifolds in $G_2$-manifolds from orbifolds.

Method constructs rigid associative submanifolds in twisted G2-manifolds.

The article develops deformation theory for ACyl associative submanifolds in ACyl G2-manifolds.

Study of associative submanifolds in Berger space SO(5)/SO(3).

Associative submanifolds $A$ in nearly parallel $G_2$-manifolds $Y$ are minimal 3-submanifolds in spin 7-manifolds with a real Killing spinor. The Riemannian cone over $Y$ has the holonomy group contained in ${\rm Spin(7)}$ and the Riemannian cone over $A$ is a Cayley submanifold. Infinitesimal deformations of associat…

Proofs and descriptions of totally geodesic submanifolds in symmetric spaces.

We study deformations of associative submanifolds $Y^3\subset M^7$ of a $G_2$ manifold $M^7$. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative subm…

We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann submanifolds. An example is given f…

Building on ideas from [DT98; DS11; Wal17; Hay17], we outline a proposal for constructing Floer homology groups associated with a G2-manifold. These groups are generated by associative submanifolds and solutions of the ADHM Seiberg-Witten equations. The construction is motivated by the analysis of various transitions w…

Let M^7 a manifold with holonomy in G_2, and Y^3 an associative submanifold with boundary in a coassociative submanifold. In [5], the authors proved that M_{X,Y}, the moduli space of its associative deformations with boundary in the fixed X, has finite virtual dimension. Using Bochner's technique, we give a vanishing t…

We give a sufficient condition for an associative submanifold in a G2-manifold to appear as the bubbling locus of a sequence of G2-instantons, related to the existence of a Fueter section of a bundle of ASD instanton moduli spaces over said submanifold.

It is well-known that in any codimension a simply connected Euclidean minimal surface has an associated one-parameter family of minimal isometric deformations. In this paper, we show that this is just a special case of the associated family to any simply connected elliptic surface for which all curvature ellipses of a …

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat…

A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y) = \mathbb{R}_{>0} \times Y$ has the holonomy group contained in ${\rm Spin(7)}$. In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$.…

By a theorem of Mclean, the deformation space of an associative submanifold Y of an integrable G_2 manifold (M,φ) can be identified with the kernel of a Dirac operator D:Ω^{0}(ν) -->Ω^{0}(ν) on the normal bundle νof Y. Here, we generalize this to the non-integrable case, and also show that the deformation space becomes…

The principal theory of this paper comprises a technique for constructing associative, coassociative and Cayley submanifolds of Euclidean space with symmetries, using first-order ordinary differential equations. Explicit examples of U(1)-invariant associative cones in R^7 and SU(2)-invariant Cayley 4-folds in R^8 are t…

Concerning the problem of classifying complete submanifolds of Euclidean space with codimension two admitting genuine isometric deformations, until now the only known examples with the maximal possible rank four are the real Kaehler minimal submanifolds classified by Dajczer-Gromoll \cite{dg3} in parametric form. These…

The paper extends the Hopf differential concept to associative submanifolds in G2-manifolds.

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C^4, as well as Lagrangian submanifolds of the near…

The squashed 7-sphere $S^{7}$ is a 7-sphere with an Einstein metric given by the canonical variation and its cone $\mathbb{R}^{8} - \{ 0 \}$ has full holonomy ${\rm Spin}(7)$. There is a canonical calibrating 4-form $Φ$ on $\mathbb{R}^{8} - \{ 0 \}$. A minimal 3-submanifold in $S^{7}$ is called associative if its cone …

We discuss various algebraic quantum structures associated to monotone Lagrangian submanifolds and we present a number of applications, computations and examples.

We revisit McLean's second variation formulas for calibrated submanifolds in exceptional geometries, and correct his formulas concerning associative submanifolds and Cayley submanifolds, using a unified treatment based on the (relative) calibration method and Harvey-Lawson's identities.

Let $M$ be a topological $G_2$-manifold. We prove that the space of infinitesimal associative deformations of a compact associative submanifold $Y$ with boundary in a coassociative submanifold $X$ is the solution space of an elliptic problem. For a connected boundary $\partial Y$ of genus $g$, the index is given by $\i…

Study on a new type of submanifolds in geometric spaces.

Study $G_2$ manifolds with symmetry, finding new submanifolds.

We investigate the local geometry of a class of Kähler submanifolds $M \subset \R^n$ which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the $(1,1)$-part (i.e. the $dz_id\bar z_j$-components) of the second fundamental form $α$, which we call the pluri-mean curvature.…

Upper bounds for Steklov eigenvalues on curved submanifolds.

Study on umbilical submanifolds in specific geometric spaces.

I present a class of examples of \CR-submanifolds of manifolds endowed with different structures, obtained as level sets of momentum maps associated to specific Hamiltonian actions.

Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces $-$ minimal isoparametric hypersurfaces and focal submanifolds. By virtue of their rich structures, we find new series of minimizing cones. They are cones over focal submanifolds and cones over suitable products among th…

We construct calibrated submanifolds of R^7 and R^8 by viewing them as total spaces of vector bundles and taking appropriate sub-bundles which are naturally defined using certain surfaces in R^4. We construct examples of associative and coassociative submanifolds of R^7 and of Cayley submanifolds of R^8. This construct…

New examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds found.

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres w…

We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampere Equations.

New inequalities for austere submanifolds established.

Given a parallel calibration $φ\in Ω^p(M)$ on a Riemannian manifold $M$, I prove that the $φ$--critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $φ$--critical submanifolds are precisely the integral manifolds of a $\mathscr{C}^\infty(M)$--linear subspace $\sP \subset Ω^p(M…

It is proved some results about existence and non existence of unit normal sections of submanifolds of the Euclidean space and sphere which associated Gauss maps are harmonic. Some applications to CMC hypersurfaces of the sphere and isoparametric submanifolds are obtained too.

In this note, we consider submanifolds of a generalized Kähler manifold that are CR-submanifolds for the two associated Hermitian structures. Then, we establish the conditions for the induced, generalized F structure to be a CRFK structure. The results extend similar conditions which we obtained for hypersurfaces in an…

We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we provide estimates for the norms of these automorphic forms and we find asymptotics of…

This paper is an overview of the idea of using contact geometry to construct invariants of immersions and embeddings. In particular, it discusses how to associate a contact manifold to any manifold and a Legendrian submanifold to an embedding or immersion. We then discuss recent work that creates invariants of immersio…

This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.

By analogy with associative and co-associative cases we introduce a class of three-dimensional non-orientable submanifolds, of almost $\mathrm{G}_2-$manifolds, modelled on planes lying in a special $\mathrm{G}_2-$orbit. An application of the Cartan-Kähler theory shows that some three-manifold can be presented in this w…

We introduce a generalization of structured manifolds as the most general Riemannian metric g associated to an affinor (tensor field of (1,1)-type) F and initiate a study of their semi-invariant submanifolds. These submanifolds are generalization of CR-submanifolds of almost complex geometry and semi-invariant submanif…

This paper classifies flat submanifolds with a special type of curvature form.

Quantum states are not entangled if submanifold is a product.

We prove the existence of canonical tubular neighbourhoods around complex submanifolds of Kähler manifolds that are adapted to both the holomorphic and symplectic structure. This is done by solving the complex Homogeneous Monge-Ampère equation on the deformation to the normal cone of the submanifold. We use this to est…