Decomposition of the Möbius energy II: The first and second variational formulas

The first and second variational formulas of the Möbius energy was calculated by several mathematicians. Direct calculation products a lot of terms which are not integrable even in the sense of Cauchy's principal value. By combining several terms appropriately, the integrability recovers, however, it is a quite hard job.

Using the decomposition which was given in the previous talk, we can calculate the variational formulas relatively easily. One can find their explicit expressions, and can show the following estimates.

Let $ \mathscr{M}_i $, $ \mathscr{G}_i $, and $ \mathscr{H}_i $ be integrands of the energy, the first variation, and the second variation of $ \mathcal{M}_i $. Assume $ \mathcal{M} (\mbox{\boldmath $ f $} ) < \infty $. (For $ \mathcal{M} $ and $ \mathcal{M}_i $, see the abstract of the previous talk.)

• If the curve and test functions are in $ H^{ 3/2 } \cap H^{1,\infty} $, then $ \mathscr{M}_i $, $ \mathscr{G}_i $, and $ \mathscr{H}_i $ are in $ L^1 $.
• If the curve and test functions are in $ C^{1,1} $, then $ \mathscr{M}_i $, $ \mathscr{G}_i $, and $ \mathscr{H}_i $ are in $ L^\infty $.
• If the curve and test functions are in $ C^2 $, then $ \mathscr{M}_i $, $ \mathscr{G}_i $, and $ \mathscr{H}_i $ are in $ C^0 $.