37 research outputs found
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 K\"ahler 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.Comment: 42 pages, v2: minor corrections, streamlined and improved exposition,
published version; Proceedings of the London Mathematical Society, Advance
Access published 17 June 201
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.Comment: 33 pages, v2: major changes for published version, mainly regarding
the twisted Dirac and d-bar operator
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 will imply bounds on all
covariant derivatives of Rm and T. (2). We show that will blow
up at a finite-time singularity, so the flow will exist as long as
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.Comment: 59 pages, v2: minor corrections and additions, accepted version for
GAF
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.Comment: Final version, 22 pages; to appear in a special volume in memory of
Paolo de Bartolomeis, Boll. UM