Deformations Of Multivalued Harmonic Functions

Abstract

We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.

Deformations of $\mathbb Z_2$-Harmonic Spinors on 3-Manifolds

“…"spinors" for the Dirac-type operator pd `d‹ q. Theorem 1.4 applies directly to the Z 2 -harmonic spinors in Example 1.7; while the deformation theory for many of these more general cases does not follow from Theorem 1.4, the differences from the present situation are expected to be primarily topological with the results following a similar analytic framework. A result similar to Theorem 1.4 for Z 2 -harmonic 1-forms follows from a result of S. Donaldson for multi-valued harmonic functions [9].…”

“…The present work grew out of attempts to develop a more robust analytic framework for these results, with an eye towards applications to gluing problems [42] and other deformation problems. As observed by S. Donaldson [9], the same analytic issues appear in many distinct geometric contexts, most of which remain unexplored [26].…”

Thomas-Yau conjecture and holomorphic curves

“…Much recent research concentrates on a more nonlinear cousin of HYM, known as the deformed Hermitian Yang-Mills equation (dHYM), expertly surveyed in [18]. Although the techniques involved in dHYM are more akin to other areas of Kähler geometry, such as Kähler-Einstein metrics and the J-equation 29 , its motivation is in part to find an improved mirror analogue of special Lagrangians, with the distant goal of constructing the Bridgeland stability on the B-side of the mirror.…”

“…Remark 5.24. Construction of special Lagrangian branched multiple covers over given special Lagrangians is currently studied by S. Donaldson [29] and S. He among others.…”

“…Thomas's suggestion is not the only possible way to achieve a GIT analogy. However, the other proposals[28][57] do not exhibit the mirror analogy in the same intuitively plausible way 29. Indeed, the recent breakthrough of Gao Chen[16] on dHYM is largely based on the methods he developed for the J-equation.…”