Total curvature and packing of knots

We establish a new relationship between total curvature of knots and crossing number. If K is a smooth knot in R³, R the cross-section radius of a uniform tube neighborhood K, L the arclength of K, and κ the total curvature of K, then

     

Total curvature and simple pursuit on domains of curvature bounded above

We show how circumradius and asymptotic behavior of curves in cat(0) and cat(K) spaces (K > 0) are controlled by growth rates of total curvature. We apply our results to pursuit and evasion games of capture type with simple pursuit motion, generalizing results that are known for convex Euclidean domains, and obtaining results that are new for convex Euclidean domains and hold on playing fields vastly more general than these.     

Total curvature of branched minimal surfaces

An intrinsic, and much simpler, proof of a generalization of Jorge and Meeks' total curvature formula for complete minimal surfaces is given.

     

Total absolute curvature and tightness of noncompact manifolds

In the first part we prove an extension of the Chern-Lashof inequality for noncompact immersed manifolds with finitely many ends. For this we give a lower bound of the total absolute curvature in terms of topological invariants of the manifold. In the second part we discuss tightness properties for such immersions. Finally, we give an upper bound for the substantial codimension.

     

Total mean curvature of positively curved hypersurfaces

In this article, we prove that every positively curved, complete non-compact hypersurface in Rn has infinite total mean curvature.     

Total curvature of manifolds in self-immersed manifolds

Chern-Lashof [3] and Kuiper [5] showed the total absolute curvature of a manifold in Euclidean space equals the mean value of the number of critical points of height functions. Teufel [10] proved that a similar result holds for the total absolute curvature of a manifold in a unit sphere. The purpose of this paper is to extend Teufel's result to a relation between the total absolute curvature of some manifolds in self-immersed manifolds and the mean value of the number of zeros of certain vector fields.     

Total absolute curvature and tight submanifolds in hyperbolic space

An integral-geometric formula for the total absolute curvatureof tight submanifolds in hyperbolic space is given. Counterexamplesto the Chern–Lashof inequality are constructed in hyperbolicspace of dimension 3.     

Total curvature of complete surfaces in hyperbolic space

We prove a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space under some assumptions on the asymptotic behavior. The result is given in terms of the measure of geodesics intersecting the surface non-trivially, and of a conformal invariant of the curve at infinity.     

Total curvature and volume of foliations on the sphere S 2

In this paper we study a curvature integral associated with a pair of orthogonal foliations on the Riemann sphere S ² and we compute the minimal value of the volume of meromorphic foliations.     

Total curvatures of geodesic spheres associated to quadratic curvature invariants

Total scalar curvatures of geodesic spheres obtained by integrating the second-order scalar invariants of the curvature tensor are investigated. The first terms in their power-series expansions are derived and these results are used to characterize the two-point homogeneous spaces among Riemannian manifolds with adapted holonomy. Dedicated to Professor L. VanheckeMathematics Subject Classification (2000) 53C25, 53C30     

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]).     

Total curvature and L 2 harmonic 1-forms on complete submanifolds in space forms

Let M ⁿ be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., ${\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}$ , in an (n + p)-dimensional simply connected space form N ^n+p (c) of constant curvature c, where |H| and |A|² are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n ≥ 3, c = 0 and ${\int_M|H|^n < \infty}$ ; (ii) n ≥ 5, c = ?1 and ${|H| < 1-\frac{2}{\sqrt n}}$ ; (iii) n ≥ 3, c = 1 and |H| is bounded, then the dimension of the space of L ² harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.     

Local minimizers and rearrangements

We are concerned witha priori estimates for functionsu which locally minimize, in the topology ofL , functionals of the Calculus of Variations. Sharp pointwise upper bounds for the spherically symmetric rearrangement ofu are proved. Such result enables us to get conditions for the boundedness ofu and estimates for ess sup¦u¦.     

Ricci curvature,radial curvature and large volume growth

In this paper, we study complete noncompact Riemannian manifolds with Ricci curvature bounded from below. When the Ricci curvature is nonnegative, we show that this kind of manifolds are diffeomorphic to a Euclidean space, by assuming an upper bound on the radial curvature and a volume growth condition of their geodesic balls. When the Ricci curvature only has a lower bound, we also prove that such a manifold is diffeomorphic to a Euclidean space if the radial curvature is bounded from below. Moreover, by assuming different conditions and applying different methods, we shall prove more results on Riemannian manifolds with large volume growth.