今井 淳

We give a corrected proof of the last theorem (Theorem 4.11) of Regularized Riesz energies of submanifolds, published in Math. Nachr. 291 (2018), no. 8–9, 1356–1373. Namely, we prove that the Riesz energy (Formula presented.) of a compact body Ω is Möbius invariant if and only if the dimension of Ω is even and (Formula presented.).

The configuration space of the mechanism of a planar robot is studied. We consider a robot which has n arms such that each arm is of length 1 + 1 and has a rotational joint in the middle, and that the endpoint of the kth arm is fixed to Re 2(k-1)pi/n i. Generically, the configuration space is diffeomorphic to an orientable closed surface. Its genus is given by a topological way and a Morse theoretical way. The homeomorphism types of it when it is singular is also given. (C) 2006 Elsevier B.V. All rights reserved.

Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of knots and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form, which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does, and that their values explode if a knot degenerates to a singular knot with double points.

We study an energy functional of knots, e(j)p (jp > 2), that is finite valued for embedded circles and takes + infinity for circles with double points. We show that for any b is-an-element-of R there are finitely many solid tori T1,..., T(m) such that any knot with e(j)p less-than-or-equal-to b can be contained in some T(i) in a good manner. Then we can show the existence of a minimizer of e(j)p in each knot type.

We define energy functionals on the space of embeddings from S1 into R3 and show the finiteness of knot types under bounded values of those functionals.