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We give a sufficient condition for the strict parabolic power concavity of the convolution in space variable of a function defined on $$\mathbb {R}^n \times (0,+\infty )$$ and a function defined on $$\mathbb {R}^n$$ . Since the strict parabolic power concavity of a function defined on $$\mathbb {R}^n \times (0,+\infty )$$ naturally implies the strict power concavity of a function defined on $$\mathbb {R}^n$$ , our sufficient condition implies the strict power concavity of the convolution of two functions defined on $$\mathbb {R}^n$$ . As applications, we show the strict parabolic power concavity and strict power concavity in space variable of the Gauss–Weierstrass integral and the Poisson integral for the upper half-space.

We give a condition for a function to produce a M\"obius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of M\"obius invariant knot energies can produce M\"obius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of M\"obius geometry., Comment: 14 pages

We show that balls, circles and 2-spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obtain the identification by the interpoint distance distribution., 9 pages, 2 figures

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 kind of converse of Pompeiu's theorem. Fix an equilateral triangle $\triangle A_0B_0C_0$, then for any triangle $\triangle ABC$ there is a unique point $P$ inside the circumcircle $\Gamma_0$ of $\triangle A_0B_0C_0$ such that a triangle with edge lengths $PA_0, PB_0$, and $PC_0$ is similar to $\triangle ABC$. It follows that an open disc inside $\Gamma_0$ can be considered as a moduli space of similarity classes of triangles. We show that it is essentially equivalent to another moduli space based on a shape function of triangles which has been used in preceding studies., Comment: 10 pages, 7 figures

Given a closed submanifold, or a compact regular domain, in Euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuation), and show that they give essentially the same result. We prove that some of these energies are invariant under MA¶bius transformations, thus giving a generalization to higher dimensions of the MA¶bius energy of knots.

We introduce several methods to define the self-inductance of a single loop as the regularization of divergent integrals obtained from Neumann and Weber formulae for the mutual inductance applied to a single loop. We give generalization of these two formulae in the sense of generalized functions and study the residues.

We study the geometry of a Euclidean triangle from a Mobius geometric point of view. It turns out that its in- and ex-centers can be constructed in a symmetric and Mobius invariant way. We relate this construction to Thurston's center of symmetry of an ideal tetrahedron in hyperbolic space and discuss some implications for the Euclidean triangle.

We generalize the Riesz potential of a compact domain in $\mathbb{R}^{m}$ by introducing a renormalization of the $r^{\alpha-m}$-potential for $\alpha\le0$. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume., Comment: Adv. Appl. Math. 48 (2012), 365--392 Figure 11 has been corrected after publication. Theorem 3.12 and the exposition of Lemma 2.15 are modified in version 8

The set of osculating circles of a given curve in S 3 forms a lightlike curve in the set of oriented circles in S 3 . We show that its “ 1 -dimensional measure” with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.

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.

We present a Gauss–Bonnet type formula for complete surfaces in n-dimensional hyperbolic space \(\mathbb{H}^{n}\) under some assumptions on their asymptotic behaviour. As in recent results for Euclidean submanifolds (see Dillen–Kuhnel [4] and Dutertre [5]), the formula involves an ideal defect, i.e., a term involving the geometry of the set of points at infinity.

The {\em minimal unfolded region} (or the {\em heart}) of a bounded subset $\Om$ in the Euclidean space is a subset of the convex hull of $\Om$ the definition of which is based on reflections in hyperplanes. It was introduced to restrict the location of the points that give extreme values of certain functions, such as potentials whose kernels are monotone functions of the distance, and solutions of differential equations to which Aleksandrov's reflection principle can be applied. %the temperature of a heat conductor with some initial-boundary condition, in which case the points are called the hot spots. We show that the minimal unfolded regions of the convex hull and parallel bodies of $\Om$ are both included in that of $\Om$., Comment: to appear in Hokkaido Math. J

We give a new proof of an old theorem by Banchoff and White 1975 that claims that the writhe of a knot is conformally invariant., Comment: 10 pages, 3 figures

We show that among tori of revolution the Clifford torus gives the minimum value of the Mobius invariant surface energy defined by Auckly and Sadun.

We study an energy functional of knots, e p j ( jp > 2), that is finite valued for embedded circles and takes +∞ for circles with double points. We show that for any b ϵ R there are finitely many solid tori T 1 ,…, T m such that any knot with e p j ⩽ b can be contained in some T i in a good manner. Then we can show the existence of a minimizer of e p j in each knot type.

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

This is a survey article on two topics. The Energy E of knots can be obtained by generalizing an electrostatic energy of charged knots in order to produce optimal knots. It turns out to be invariant under Moebius transformations. We show that it can be expressed in terms of the infinitesimal cross ratio, which is a conformal invariant of a pair of 1-jets, and give two kinds of interpretations of the real part of the infinitesimal cross ratio., Comment: This is the version published by Geometry & Topology Monographs on 19 March 2008

We define energy functionals on the space of embeddings from S1 into R 3 and show the finiteness of knot types under bounded values of those functionals.

Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot 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., Comment: 66 pages, 25 figures, to appear in J. Institut Math. Jussieu

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’.