Properties
- The Hitchin functional is analogous for six-manifold to the Yang-Mills functional for the four-manifolds.
- The Hitchin functional is manifestly invariant under the action of the group of orientation-preserving diffeomorphisms.
- Theorem. Suppose that is a three-dimensional complex manifold and is the real part of a non-vanishing holomorphic 3-form, then is a critical point of the functional restricted to the cohomology class . Conversely, if is a critical point of the functional in a given comohology class and, then defines the structure of a complex manifold, such that is the real part of a non-vanishing holomorphic 3-form on .
- The proof of the theorem in Hitchin's articles Hitchin (2000) and Hitchin (2001) is relatively straightforward. The power of this concept is in the converse statement: if the exact form is known, we only have to look at its critical points to find the possible complex structures.
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