arXiv subjects

Jason D. Lotay

Publications and source records attributed to Jason D. Lotay.

36 records · Page 2Linked to original sources

arXiv2018

Geometric Flows of G_2 Structures

Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the context of G_2 geometry, there are several geometric flows which arise. Each flow provides a potential means to study the geometry and topology associated with a given class of G_2 structures. We will introduce these flows, and describe some of the key known results and open problems in the field.

math.DG↗

arXiv2018

Calibrated Submanifolds

We provide an introduction to the theory of calibrated submanifolds through the key examples related with special holonomy. We focus on calibrated geometry in Calabi-Yau, G$_2$ and Spin(7) manifolds, and describe fundamental results and techniques in the field.

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arXiv2018

From minimal Lagrangian to J-minimal submanifolds: persistence and uniqueness

Given a minimal Lagrangian submanifold L in a negative Kaehler--Einstein manifold M, we show that any small Kaehler--Einstein perturbation of M induces a deformation of L which is minimal Lagrangian with respect to the new structure. This provides a new source of examples of minimal Lagrangians. More generally, the same is true for the larger class of totally real J-minimal submanifolds in Kaehler manifolds with negative definite Ricci curvature.

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arXiv2018

Consequences of strong stability of minimal submanifolds

In this note we show that the recent dynamical stability result for small $C^1$-perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang directly extends to the enhanced Brakke flows of Ilmanen. We illustrate applications of this result, including a local uniqueness statement for strongly stable minimal submanifolds amongst stationary varifolds, and a mechanism to flow through some singularities of Lagrangian mean curvature flow which are proved to occur by Neves.

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arXiv2017

$SU(2)^2$-invariant $G_2$-instantons

We initiate the systematic study of $G_2$-instantons with $SU(2)^2$-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on $\mathbb{R}^4\times S^3$ with its two explicitly known distinct holonomy $G_2$ metrics, which have different volume growths at infinity, exhibiting the different behaviour of instantons in these settings. We also give an explicit example of sequences of $G_2$-instantons where "bubbling" and "removable singularity" phenomena occur in the limit.

math.DG↗

arXiv2017

From Lagrangian to totally real geometry: coupled flows and calibrations

We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kähler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric flow in terms of the ambient canonical bundle, (ii) a modified volume functional which takes into account the totally real condition. We discuss short-time existence for our flow and show it couples well with the Streets-Tian symplectic curvature flow for almost Kähler manifolds. We also discuss possible applications to Lagrangian submanifolds and calibrated geometry.

math.DG↗

arXiv2017

Complexified diffeomorphism groups, totally real submanifolds and Kähler-Einstein geometry

Let (M,J) be an almost complex manifold. We show that the infinite-dimensional space Tau of totally real submanifolds in M carries a natural connection. This induces a canonical notion of geodesics in Tau and a corresponding definition of when a functional, defined on Tau, is convex. Geodesics in Tau can be expressed in terms of families of J-holomorphic curves in M; we prove a uniqueness result and study their existence. When M is Kähler we define a canonical functional on Tau; it is convex if M has non-positive Ricci curvature. Our construction is formally analogous to the notion of geodesics and the Mabuchi functional on the space of Kähler potentials, as studied by Donaldson, Fujiki and Semmes. Motivated by this analogy, we discuss possible applications of our theory to the study of minimal Lagrangians in negative Kähler-Einstein manifolds.

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arXiv2017

Stability of torsion-free G_2 structures along the Laplacian flow

We prove that torsion-free G_2 structures are (weakly) dynamically stable along the Laplacian flow for closed G_2 structures. More precisely, given a torsion-free G_2 structure $φ$ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to $φ$ will converge to a torsion-free G_2 structure which is in the orbit of $φ$ under diffeomorphisms isotopic to the identity.

math.DG↗

arXiv2017

The space of hyperkähler metrics on a 4-manifold with boundary

Let X be a compact 4-manifold with boundary. We study the space of hyperkähler triples on X, modulo diffeomorphisms which are the identity on the boundary. We prove that this moduli space is a smooth infinite-dimensional manifold and describe the tangent space in terms of triples of closed anti-self-dual 2-forms. We also explore the corresponding boundary value problem: a hyperkähler triple restricts to a closed framing of the bundle of 2-forms on the boundary; we identify the infinitesimal deformations of this closed framing that can be filled in to hyperkähler deformations of the original triple. Finally we study explicit examples coming from gravitational instantons with isometric actions of SU(2).

math.DG↗

arXiv2017

Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness

We develop foundational theory for the Laplacian flow for closed G_2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on $Λ(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$ will imply bounds on all covariant derivatives of Rm and T. (2). We show that $Λ(x,t)$ will blow up at a finite-time singularity, so the flow will exist as long as $Λ(x,t)$ remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2). (5). Finally, we study compact soliton solutions of the Laplacian flow.

math.DG↗

arXiv2017

Laplacian flow for closed G_2 structures: real analyticity

Let $φ(t), t\in [0,T]$ be a smooth solution to the Laplacian flow for closed G_2 structures on a compact 7-manifold $M$. We show that for each fixed positive time $t\in (0,T]$, $(M,φ(t),g(t))$ is real analytic, where $g(t)$ is the metric induced by $φ(t)$. Consequently, any Laplacian soliton is real analytic and we obtain unique continuation results for the flow.

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arXiv2015

Desingularization of Coassociative 4-folds with Conical Singularities: Obstructions and Applications

We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for deformations of conically singular coassociative 4-folds, and thus relate them to the stability of the singularities. We use our results to describe the relationship between moduli spaces of coassociative 4-folds with conical singularities and those of their desingularizations. We also apply our theory in examples, including to the known conically singular coassociative 4-folds in compact holonomy G_2 manifolds.

math.DG↗

arXiv2012

Stability of Coassociative Conical Singularities

We study the stability of coassociative 4-folds with conical singularities under perturbations of the ambient G_2 structure by defining an integer invariant of a coassociative cone which we call the stability index. The stability index of a coassociative cone is determined by the spectrum of the curl operator acting on its link. We explicitly calculate the stability index for cones on group orbits. We also describe the stability index for cones fibered by 2-planes over algebraic curves using the degree and genus of the curve and the spectrum of the Laplacian on the link. Finally we apply our results to construct the first known examples of coassociative 4-folds with conical singularities in compact manifolds with G_2 holonomy.

math.DG↗

arXiv2012

Uniqueness of Lagrangian Self-Expanders

We show that zero-Maslov class Lagrangian self-expanders in C^n which are asymptotic to a pair of planes intersecting transversely are locally unique if n>2 and unique if n=2.

math.DG↗

arXiv2012

Associative Submanifolds of the 7-Sphere

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 nearly Kähler 6-sphere. By classifying the associative group orbits, we exhibit the first known explicit example of an associative 3-fold in S^7 which does not arise from other geometries. We then study associative 3-folds satisfying the curvature constraint known as Chen's equality, which is equivalent to a natural pointwise condition on the second fundamental form, and describe them using a new family of pseudoholomorphic curves in the Grassmannian of 2-planes in R^8 and isotropic minimal surfaces in S^6. We also prove that associative 3-folds which are ruled by geodesic circles, like minimal surfaces in space forms, admit families of local isometric deformations. Finally, we construct associative 3-folds satisfying Chen's equality which have an S^1-family of global isometric deformations using harmonic 2-spheres in S^6.

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arXiv2010

Ruled Lagrangian Submanifolds of the 6-Sphere

This article sets out to serve a dual purpose. On the one hand, we give an explicit description of the Lagrangian submanifolds of the nearly Kaehler 6-sphere which are ruled by circles of constant radius using Weierstrass formulae. On the other, we recognise all previous known examples of these Lagrangians as being ruled by such circles. Therefore, we describe all families of Lagrangians in the 6-sphere whose second fundamental form satisfies natural pointwise conditions: so-called second order families.

math.DG↗

arXiv2010

Asymptotically Conical Associative 3-folds

Given an associative 3-fold in R^7 which is asymptotically conical with generic rate less than 1, we show that its moduli space of deformations is locally homeomorphic to the kernel of a smooth map between smooth manifolds. Moreover, the virtual dimension of the moduli space is computed and shown to be non-negative for rates greater than -1, whereas the associative 3-fold is expected to be isolated for rates less than or equal to -1.

math.DG↗

arXiv2008

Deformations of Compact Coassociative 4-folds with Boundary

Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G_2 structure given by a `positive' differential 3-form, sometimes called G_2-form. Assuming that a G_2-form on M is closed, we study deformations of a compact coassociative submanifold N with boundary contained in fixed, codimension 1 submanifold S of M with a compatible Hermitian symplectic structure. We show that `small' coassociative deformations of N with special Lagrangian boundary in S are unobstructed and form a smooth moduli space of finite dimension not greater than the first Betti number of the boundary of N. It is also shown that N is `stable' under small deformations of the closed G_2-form on the ambient 7-manifold M. The results can be compared to those for special Lagrangian submanifolds of Calabi--Yau manifolds proved by A.Butscher in math.DG/0110052.

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