arXiv subjects
Publications and source records attributed to Ryohei Chihara.
We sketch an application of proximal algorithms to the deformation of de Rham currents into cycles, which is presented as a convex optimization problem. Emphasis is placed on the use of total variation denoising for differential forms, specifically in constructing calibrated cycles in calibrated manifolds.
We study a bordism relation for stable 3-forms on a 6-manifold, which is a binary relation on the set of closed $SL(3;\mathbb{C})$-structures on a 6-manifold via closed $G_2$-structures. Under $SO(3)$-symmetry and a co-associative condition the relation is reduced to a relation for geometric structures on a 3-manifold. Under these conditions we prove that the bordism relation is irreflexive and that the relation induces a more rigid one.
We study special Lagrangian fibrations of $\mathrm{SU}(3)$-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group $G$, we decompose such $\mathrm{SU}(3)$-structures into triples of solder 1-forms, connection 1-forms and equivariant $3\times3$ positive-definite symmetric matrix-valued functions on principal $G$-bundles over 3-manifolds. As applications, we describe regular parts of $G_2$-manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of $G=\mathrm{T}^3$ and $\mathrm{SO}(3)$.
In this paper we study a Hamiltonian function on the cotangent bundle of the space of Riemannian metrics on a 3-manifold $M$ and prove the orbits of the constrained Hamiltonian dynamical system correspond to $G_2$-manifolds foliated by hypersurfaces diffeomorphic to $M\times \mathrm{SO}(3)$.