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DOES FINITE KNOT ENERGY LEAD TO DIFFERENTIABILITY?
Abstract
In this article, we raise the question if curves of finite (j, p)-knot energy introduced by O'Hara are at least pointwise differentiable. If we exclude the highly singular range (j - 2)p ≥ 1, the answer is no for jp ≤ 2 and yes for jp > 2. In the first case, which also contains the most prominent example of the Möbius energy(j = 2, p = 1) investigated by Freedman, He and Wang, we construct counterexamples. For jp > 2, we prove that finite-energy curves have in fact a Hölder continuous tangent with Hölder exponent ½(jp - 2)/(p + 2). Thus, we obtain a complete picture as to what extent the (j, p)-energy has self-avoidance and regularizing effects for (j, p) ∈ (0, ∞) × (0, ∞). We provide results for both closed and open curves.
AMSC: Primary 53A04, Secondary 26A27, Secondary 57M25
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