arXiv subjects

Jesse Madnick

Publications and source records attributed to Jesse Madnick.

21 records · Page 2Linked to original sources

arXiv2020

Associative Submanifolds of the Berger Space

We study associative submanifolds of the Berger space SO(5)/SO(3) endowed with its homogeneous nearly-parallel G2-structure. We focus on two geometrically interesting classes: the ruled associatives, and the associatives with special Gauss map. We show that the associative submanifolds ruled by a certain special type of geodesic are in correspondence with pseudo-holomorphic curves in $Gr_2^+(TS^4).$ Using this correspondence, together with a theorem of Bryant on superminimal surfaces in $S^4,$ we prove the existence of infinitely many topological types of compact immersed associative 3-folds in SO(5)/SO(3). An associative submanifold of the Berger space is said to have special Gauss map if its tangent spaces have non-trivial SO(3)-stabiliser. We classify the associative submanifolds with special Gauss map in the cases where the stabiliser contains an element of order greater than 2. In particular, we find several homogeneous examples of this type.

math.DG↗

arXiv2019

The Mean Curvature of First-Order Submanifolds in Exceptional Geometries with Torsion

We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those G2-structures (resp., Spin(7)-structures) for which every associative 3-fold (resp. coassociative 4-fold, Cayley 4-fold) is a minimal submanifold. In the process, we obtain new obstructions to the local existence of coassociative 4-folds in G2-structures with torsion.

math.DG↗

arXiv2018

Nearly-Kahler 6-Manifolds of Cohomogeneity Two: Local Theory

We study nearly-Kahler 6-manifolds equipped with a cohomogeneity-two Lie group action for which the principal orbits are coisotropic. If the metric is complete, then we show that this last condition is automatically satisfied, and both the acting Lie group and the principal orbits are finite quotients of $S^3 \times S^1$. We partition the class of such nearly-Ka}hler structures into three types (called I, II, III) and prove a local existence and generality result for each type. Metrics of Types I and II are shown to be incomplete. We also derive a quasilinear elliptic PDE system on the 2-dimensional orbit space which nearly-Kahler structures of Type I must satisfy. Finally, we remark on a relatively simple one-parameter family of Type III structures that turn out to be incomplete metrics that are cohomogeneity-one under the action of a larger group.

math.DG↗