One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula

In [10] one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane \mathbb E²{{\mathbb E}^2} were obtained. Moreover, the relations between the complex velocities of one-parameter motion in the complex plane were provided by [10]. One-parameter planar homothetic motion was defined in the complex plane, [9]. In this paper, analogous to homothetic motion in the complex plane given by [9], one-parameter planar homothetic motion is defined in the hyperbolic plane. Some characteristic properties about the velocity vectors, the acceleration vectors and the pole curves are given. Moreover, in the case of homothetic scale h identically equal to 1, the results given in [15] are obtained as a special case. In addition, three hyperbolic planes, of which two are moving and the other one is fixed, are taken into consideration and a canonical relative system for one-parameter planar hyperbolic homothetic motion is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, is obtained with the help of this relative system.     

Curvature Measures and Random Sets,I

The paper deals with signed curvature measures as introduced by Federer for sets with positive reach. An integral representation and a local Steiner formula for these measures are given. The main result is the additive extension of the curvature measures to locally finite unions of compatible sets with positive reach. Within this comprehensive class of subsets of R^d a generalized Steiner polynomial (local version) and section theorems (principal kinematic formula, Crofton formula) for the curvature measures are derived.     

Curvature measures of convex bodies

Summary The curvature measures, introduced by Federer for the sets of positive reach, are investigated in the special case of convex bodies. This restriction yields additional results. Among them are:(5.1), an integral-geometric interpretation of the curvature measure of order m, showing that it measures, in a certain sense, the affine subspaces of codimension m+1 which touch the convex body;(6.1), an axiomatic characterization of the (linear combinations of) curvature measures similar to Hadwiger's characterization of the quermassintegrals of convex bodies;(8.1), the determination of the support of the curvature measure of order m, which turns out to be the closure of the m-skeleton of the convex body. Moreover we give, for the case of convex bodies, a new and comparatively short proof of an integral-geometric kinematic formula for curvature measures. Entrata in Redazione il 14 dicembre 1976.     

Curvatures and Currents for Unions of Sets with Positive Reach, II

A class of subsets of ^d which can berepresented as locally finite unions of sets with positive reach isconsidered. It plays a role in PDE's on manifolds with singularities.For such a set, the unit normal cycle (determining the d – 1curvature measures) is introduced as a (d – 1)-currentsupported by the unit normal bundle and its properties are established.It is shown that, under mild additional assumptions, the unit normalcycle (and, hence, also the curvature measures) of such a set can beapproximated by that of a close parallel body or, alternatively, by themirror image of that of the closure of the complement of the parallelbody (which has positive reach). Finally, the mixed curvature measuresof two sets of this class are introduced and a translative integralgeometric formula for curvature measures is proved.     

Total absolute curvature of polyhedral manifolds with boundary in E n

N. H. Kuiper has generalized the notion of total absolute curvature for compact polyhedra in euclidean space by considering the critical points of all height functions (cf. [12]). On the other hand in the case of compact smooth manifolds with boundary in E ⁿ there is a certain relation between the total absolute curvatures of the total space, the interior and the boundary (cf. [9]). In this note we show an analogous relation in the case of compact polyhedral manifolds with boundary leading to theorems of the Chern/Lashof type (cf. [3], [7]).     

Hessian measures of semi-convex functions and applications to support measures of convex bodies

This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of σ-finite Hausdorff measure. The results which we obtain in the setting of the theory of convex bodies (sets of positive reach) are achieved as applications of various new results on Hessian measures of convex (semi-convex) functions. Among these are a Crofton formula, results on the absolute continuity of Hessian measures, and a duality theorem which relates the Hessian measures of a convex function to those of the conjugate function. In particular, it turns out that curvature and surface area measures of a convex body K are the Hessian measures of special functions, namely the distance function and the support function of K. Received: 15 July 1999     

Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.     

Generalized Interpolation in Bergman Spaces and Extremal Functions

In a recent paper A. Schuster and K. Seip [SchS] have characterized interpolating sequences for Bergman spaces in terms of extremal functions (or canonical divisors). As these are natural analogues in Bergman spaces of Blaschke products, this yields a Carleson type condition for interpolation. We intend to generalize this idea to generalized free interpolation in weighted Bergman spaces B^{p, α} as was done by V. Vasyunin [Va1] and N. Nikolski [Ni1] (cf.also [Ha2]) in the case of Hardy spaces. In particular we get a strong necessary condition for free interpolation in B^{p, α} on zero–sets of B^{p, α}–functions that in the special case of finite unions of B^{p, α}–interpolating sequences turns out to be also sufficient.     

Singular sets of convex bodies and surfaces with generalized curvatures

In this paper we consider generalized surfaces with curvature measures and we study the properties of those k-dimensional subsets Σ ^k of such surfaces where the curvatures have positive density with respect to k-dimensional Hausdorff measure. Special attention is given to boundaries of convex bodies inR ³. We introduce a class of convex sets whose curvatures live only on integer dimension sets. For such convex sets we consider integral functionals depending on the curvature and the area ofK and on the curvature andH ^k of Σ ^k .     

The classification of homotopy classes of bounded curvature paths

A bounded curvature path is a continuously differentiable piecewise C² path with bounded absolute curvature that connects two points in the tangent bundle of a surface. In this note we give necessary and sufficient conditions for two bounded curvature paths, defined in the Euclidean plane, to be in the same connected component while keeping the curvature bounded at every stage of the deformation. Following our work in [3], [2] and [4] this work finishes a program started by Lester Dubins in [6] in 1961.     

On the fundamental groups of manifolds with integral curvature bounds

In this paper, we give an upper bound on the growth of π₁(M) for a class of manifolds with integral Ricci curvature bounds. This generalizes the main theorem of [8] to the case where the negative part of Ricci curvature is small in an averaged L¹- sense.Received: 19 July 2004     

A local Steiner–type formula for general closed sets and applications

We introduce support (curvature) measures of an arbitrary closed set A in ^d and establish a local Steiner–type formula for the localized parallel volume of A. We derive some of the basic properties of these support measures and explore how they are related to the curvature measures available in the literature. Then we use the support measures in analysing contact distributions of stationary random closed sets, with a particular emphasis on the Boolean model with general compact particles. Mathematics Subject Classification (2000): 53C65, 28A75, 52A22, 60D05; 52A20, 60G57, 60G55, 28A80.     

Prescribing scalar curvature on Sn and related problems,part II: Existence and compactness

This is a sequel to [30], which studies the prescribing scalar curvature problem on Sⁿ. First we present some existence and compactness results for n = 4. The existence result extends that of Bahri and Coron [4], Benayed, Chen, Chtioui, and Hammami [6], and Zhang [39]. The compactness results are new and optimal. In addition, we give a counting formula of all solutions. This counting formula, together with the compactness results, completely describes when and where blowups occur. It follows from our results that solutions to the problem may have multiple blowup points. This phenomena is new and very different from the lower-dimensional cases n = 2, 3. Next we study the problem for n ≥ 3. Some existence and compactness results have been given in [30] when the order of flatness at critical points of the prescribed scalar curvature functions K(x) is β ϵ (n − 2, n). The key point there is that for the class of K mentioned above we have completed L^∞ apriori estimates for solutions of the prescribing scalar curvature problem. Here we demonstrate that when the order of flatness at critical points of K(x) is β = n − 2, the L^∞ estimates for solutions fail in general. In fact, two or more blowup points occur. On the other hand, we provide some existence and compactness results when the order of flatness at critical points of K(x) is β ϵ [n − 2,n). With this result, we can easily deduce that C^∞ scalar curvature functions are dense in C^1,α (0 < α < 1) norm among positive functions, although this is generally not true in the C² norm. We also give a simpler proof to a Sobolev-Aubin-type inequality established in [16]. Some of the results in this paper as well as that of [30] have been announced in [29]. © 1996 John Wiley & Sons, Inc.     

On the Central Limit Theorem for Nonuniform φ-Mixing Random Fields

The partial-sum processes, indexed by sets, of a stationary nonuniform -mixing random field on the d-dimensional integer lattice are considered. A moment inequality is given from which the convergence of the finite-dimensional distributions to a Brownian motion on the Borel subsets of [0, 1] ^d is obtained. A Uniform CLT is proved for classes of sets with a metric entropy restriction and applied to certain Gibbs fields. This extends some results of Chen⁽⁵⁾ for rectangles. In this case and when the variables are bounded a simpler proof of the uniform CLT is given.     

Normal Currents: Structure, Duality Pairings and div–curl Lemmas

The paper gives a decomposition of a general normal r-dimensional current [5] into the sum of three measures of which the first is an r-dimensional rectifiable measure, the second is the Cantor part of the current, and the third is Lebesgue absolutely continuous. This is analogous to the well-known decomposition of the derivative of a function of bounded variation into the jump, Cantor, and absolutely continuous parts; in fact the last is a special case of the result for (n–1)-dimensional normal currents. Further, Whitney’s cap product [15] is recast in the language of the approach to flat chains by Federer [5] and a special case (viz., currents of dimension n – 1) is shown to be closely related to the measure-valued duality pairings between vector measures with curl a measure and L^∞ vectorfields with L^∞ divergence as established by Anzellotti [2] and Kohn & Témam [6]. Finally, the cap product is shown to be jointly weak* continuous in the two factors of the product in a way similar to the compensated compactness theory; in the cases of (n – 1)-dimensional objects this reduces to results closely related to the div–curl lemmas of the standard compensated compactness theory. Received: June 2007     

Minimal total absolute curvature for immersions

Immersions or maps of closed manifolds in Euclidean space, of minimal absolute total curvature are called tight in this paper. (They were called convex in [25].) After the definition in Chapter 1, many examples in Chapter 2, and some special topics in Chapter 3, we prove in Chapter 4 that topological tight immersions ofn-spheres are only of the expected type, namely embeddings onto the boundary of a convexn+1-dimensional body. This generalises a theorem of Chern and Lashof in the smooth case. In Chapter 5 we show that many manifolds exist that have no tight smooth immersion in any Euclidean space.This research was partially supported by National Science Foundation grant GP-7952X1.     

Anwendungen der differentialtopologischen Berechnung der Totalen Krümmung und Totalen Absolutkrümmung in der sphärischen Differentialgeometrie

In a previous article [16] we have shown how the total absolute (Lipschitz-Killing) curvature of the immersion f:MSⁿ of a compact manifold into a sphere can be computed in a differential topological manner as the mean-value of the number of critical points of certain level-functions. (And similar the total curvature.) (comp. [12], prob. 15) Now we consider the gradient vector field of the level-functions and achieve a relation between the total curvature and the Euler characteristic, of the manifold, which can be sharpened in some cases to inequalities. Moreover it leads to the formula of Allendoerfer-Weil for compact n-dim. submanifolds of the sphere Sⁿ.     

Curvatures and currents for unions of sets with positive reach

For locally finite unions of sets with positive reach in R ^d, generalized unit normal bundles are introduced in support of a certain set additive index function. Given an appropriate orientation to the normal bundle, signed curvature measures may be defined by means of associated locally rectifiable currents (with index function as multiplicity) and specially chosen differential forms. In the case of regular sets this is shown to be equivalent to well-known classical concepts via former results. The present approach leads to unified methods in proving integral-geometric relations. Some of them are stated in this paper.     

An Integral Formula for Lipschitz-Killing Curvature and the Critical Points of Height Functions

We have established (see Shiohama and Xu in J. Geom. Anal. 7:377–386, 1997; Lemma) an integral formula on the absolute Lipschitz-Killing curvature and critical points of height functions of an isometrically immersed compact Riemannian n-manifold into R ^n+q . Making use of this formula, we prove a topological sphere theorem and a differentiable sphere theorem for hypersurfaces with bounded L ^n/2 Ricci curvature norm in R ⁿ⁺¹. We show that the theorems of Gauss-Bonnet-Chern, Chern-Lashof and the Willmore inequality are all its consequences.