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We show that the regularized Riesz $\a$-energy for closed submanifolds $M$ in $\RR^n$ blows up as $M$ degenerates to have double points if $\a\le-2\dim M$. This gives theoretical foundation of numerical experiments to evolve surfaces to decrease the energy which have been carried out since 90's., 14 pages, two figures

We introduce a Mobius invariant energy associated to planar domains, as well as a generalization to space curves. This generalization is a Mobius version of Banchoff-Pohl's notion of area enclosed by a space curve. A relation with Gauss-Bonnet theorems for complete surfaces in hyperbolic space is also described.

A two-component link produces a torus as the product of the component knots in a two-point configuration space of a three-sphere. This space can be identified with a cotangent bundle and also with an indefinite Grassmannian. We show that the integration of the absolute value of the canonical symplectic form is equal to the area of the torus with respect to the pseudo-Riemannian structure, and that it attains the minimum only at the "best" Hopf links., to appear in Tohoku Math. J

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 $k$-th arm is fixed to $Re^{\frac{2(k-1)\pi}ni}$. 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., Comment: 33 pages, 41 figures

IN THIS paper. we define a real valued functional. the unuryr E. which is well-behaved for embedded circles in the 3 dimensional Euclidean space, and which blows up for curves with self-intersections. Let EiB be the sum of the energy E and the total squared curvature functional. We show that for any real number z. there are only finitely many ambient isotopy clusscs of embeddinps (i.e. knor t)‘pc’s) with the value of E‘iR not greater than x. There have been studied the total curvature (Fary [I]. Fenchel [7]. Milnor [5]), the total squared curvature (Lanpcr ilnd Singer [-I]), and the Gauss integral of the linking numhcr for a single curve, which. with the total torsion, leads to the notion of the self linking number (Pohl [7]) as functionals on the space of closed curves in R.’ with suitable conditions. Hut these functionals do not have the above properties. They do not blow up for curves with self-intcrscctions. and we can not in general show the finiteness of knot types by them. though we can distinguish the trivial knot from non-trivial knots by the total curvature. and hence. by the total squared curvature ([I], [S]). “Energy” of polygonal knots which is something like electrostatic energy was studied by Fukuhara [3], and “energy” of geodesic links in S’ which is defined by the principal angles was studied by Sakuma [U]. This work was motivated by [8]. in #I. we define the energy E, show the continuity of E, give a lower bound of E and the formulation for E by the double integral, and state a fundamental property of E. In $2, we show the finiteness of the knot types under the bounded value of EAB. Throughout this paper, we always consider the embeddings from S * into R’ of class Cz such that the norm of the derivative is always one. We use the notation 1.1 for the standard norm of R’.