BIMSA >
Differential Geometry Seminar
A special class of k-harmonic maps inducing calibrated fibrations.
A special class of k-harmonic maps inducing calibrated fibrations.
Organizers
Speaker
Time
Tuesday, September 24, 2024 3:00 PM - 4:00 PM
Venue
A7-201
Online
Zoom 518 868 7656
(BIMSA)
Abstract
We consider two special classes of k-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map u:(L^k,g)→(M^n,h) where k≤n and α is a calibration k-form on M. Away from the critical set, the image is an α-calibrated submanifold of M. These were previously studied by Cheng-Karigiannis-Madnick when α was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map u:(M^n,h)→(L^k,g) where n≥k and α is a calibration (n−k)-form on M. Away from the critical set, the fibres u^{−1}{u(x)} are α-calibrated submanifolds of M. We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where (M,h) are the Bryant-Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger-Yau-Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the G_2 version by Gukov-Yau-Zaslow in terms of coassociative fibrations; and we present several open questions for future study.