Frank Morgan

In Section 3 of his article on “Crystals, Proteins, Stability and Isoperimetry” in the April 2011 Bulletin of the American Mathematical Society, Misha Gromov suggests that a complex subvariety S of a Kähler manifold X minimizes a curvature energy defined as the volume of the unit tangent bundle S′ in the Grassmannian bundle X′ of X. Here we note that this holds for complex curves in the following sense:

Proposition. Let S be a complex curve in Cⁿ. Then compact portions of S minimize curvature energy among surfaces (rectifiable currents) with the same boundary and the same tangent planes (almost everywhere) along the boundary.

Proof. The Grassmannian bundle is a Kähler manifold, and S′ is a complex curve, hence volume minimizing for given homology class and boundary, which corresponds to boundary and tangent planes along the boundary for S.

For example {x²=y³} in C², with that interesting singularity at the origin. I don’t know what condition on a more general Kähler manifold X guarantees that X′ is a Kähler manifold.

Incidentally, there is a recent arXiv post on “Willmore minimizers with prescribed isoperimetric ratio” by Johannes Schygulla.