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.
Read more about this topic: Hitchin Functional
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)