Constructing Associative 3-folds by Evolution Equations

Details Constructing Associative 3-folds by Evolution Equations Constructing Associative 3-folds by Evolution Equations

2004-01-13

arxiv.org/abs/math/0401123v2

Communications in Analysis and Geometry, Volume 13, Number 5, 999-1037, 2005

Mathematics

Differential Geometry

43 pages, LaTeX; minor corrections, mainly typos, and slight changes in presentation, including added references

Scientific paper

This paper gives two methods for constructing associative 3-folds in R^7, based around the fundamental idea of evolution equations, and uses these methods to construct examples of these geometric objects. The paper is a generalisation of the work by Joyce in math.DG/0008021, math.DG/0008155, math.DG/0010036 and math.DG/0012060 on special Lagrangian 3-folds in C^3. The two methods described involve the use of an affine evolution equation with affine evolution data and the area of ruled submanifolds. We first give a derivation of an evolution equation for associative 3-folds from which we derive an affine evolution equation using affine evolution data. We then use this on an example of such data to construct a 14-dimensional family of associative 3-folds. One of the main result of the paper is then an explicit solution of the system of differential equations generated in a particular case to give a 12-dimensional family of associative 3-folds. We also find that there is a straightforward condition that ensures that the associative 3-folds constructed are closed and diffeomorphic to S^1xR^2, rather than R^3. In the final section we define ruled associative 3-folds and derive an evolution equation for them. This then allows us to characterise a family of ruled associative 3-folds using two real analytic maps that must satisfy two partial differential equations. We finish by giving a means of constructing ruled associative 3-folds M from r-oriented two-sided associative cones M_0 such that M is asymptotically conical to M_0 with order O(r^{-1}).

Affiliated with

Also associated with

No associations

Rating

Constructing Associative 3-folds by Evolution Equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Constructing Associative 3-folds by Evolution Equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Constructing Associative 3-folds by Evolution Equations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-266668