Volume 21, Issue 6

Hopf's theorem has been recently extended to compact genus zero surfaces with constant mean curvature H in a product space , where is a surface with constant Gaussian curvature k ≠ 0 [Abresch, Rosenberg, Acta Math. 193: 141–174, 2004]. It also has been observed that, rather than H = const., it suffices to assume that the differential dH of H is appropriately bounded [Alencar, do Carmo, Tribuzy, Analysis Geometry 15: 283–298, 2007]. Here, we consider the case of simply-connected open surfaces with boundary in such that dH is appropriately bounded and certain conditions on the boundary are satisfied, and show that such surfaces can all be described.

Steinberg's theorem on the coinvariant algebra of a complex representation of a finite group G says that is a Poincaré duality algebra if and only if the invariant algebra is a polynomial algebra. The extension of this to the nonmodular case has been achieved in stages, the final result being obtained by W.G. Dwyer and C.W. Wilkerson. We show that the main module theoretic tool they use extends to the following characteristic free result: If 𝔽[ V ] G is a Poincaré duality algebra of formal dimension d , then 𝔽[ V ] G is a polynomial algebra if and only if contains a nonzero element of degree – d . In the nonmodular case an easy transfer argument then recovers their extension of Steinberg's theorem by means of some representation theory. Combined with some new results concerning the Δ operators of Demazure, our characteristic free result yields the following for reflection groups: A reflection group G for which 𝔽[ V ] G is a Poincaré duality algebra in which the trivial G -representation 1 G occurs only once as a subrepresentation has a polynomial algebra for its invariant algebra 𝔽[ V ] G .

Let us consider two closed curves ℳ, of class C 1 and two functions of class C 1 , called measuring functions. The natural pseudo-distance d between the pairs (ℳ, φ ), (, ψ ) is defined as the infimum of , as ƒ varies in the set of all homeomorphisms from ℳ onto . The problem of finding the possible values for d naturally arises. In this paper we prove that under appropriate hypotheses the natural pseudo-distance equals either | c 1 – c 2 | or , where c 1 and c 2 are two suitable critical values of the measuring functions. This equality shows that the relations between the natural pseudo-distance and the critical values of the measuring functions previously obtained in higher dimensions can be made stronger in the particular case of closed curves. Moreover, the examples we give in this paper show that our result cannot be further improved, and therefore it completely solves the problem of determining the possible values for d in the 1-dimensional case.

We investigate the ideal semigroup and the ideal class semigroup built by the fractional ideals of an ideal system on a monoid or on a domain. We provide criteria for these semigroups to be Clifford semigroups or Boolean semigroups. In particular, we consider the case of valuation monoids (domains) and of Prüfer-like monoids (domains). By the way, we prove that a monoid (domain) is of Krull type if every locally principal ideal is finite.

We study the growth rate of L p norms of eigenfunctions of the Laplace-Beltrami operator restricted to submanifolds of compact C ∞ Riemannian manifolds. The spectral projection operators can be expressed as oscillatory integral operators, so the question reduces to oscillatory integral operator norm estimates. We study the relationship between the extrinsic geometry of these submanifolds and the canonical relations associated to these oscillatory integral operators.

We give a nontrivial estimate of the central value of the generalized multiple sine function by a new integral expression of its logarithm. Moreover, when all the periods coincide with 1, we show relations between the central values of the generalized multiple sine functions and the special values of the Riemann zeta function.

For a symmetric stable process we consider a transform of its transition semigroup by a non-local Feynman-Kac functional. We prove that if the Feynman-Kac functional belongs to a certain class, then the spectral bound of the transformed semigroup on L p (ℝ d ) is independent of p .

Some results about products of pairwise mutually permutable subgroups are presented in this paper. It is shown that this kind of products behaves well with respect to some well-known classes of groups. For instance, we show that all factors have only simple chief factors if the product has this property. This is necessary but not sufficient: we need that the factors belong to the subclass of PST-groups to make sure that the product has only simple chief factors (see Theorems 5 and 6).

For a finite dimensional real vector space V with inner product, let F ( V ) be the block structure space, in the sense of surgery theory, of the projective space of V . Continuing a program launched in [Macko, Trans. Amer. Math. Soc. 359: 349–383, 2007], we investigate F as a functor on vector spaces with inner product, relying on functor calculus ideas. It was shown in [Macko, Trans. Amer. Math. Soc. 359: 349–383, 2007] that F agrees with its first Taylor approximation T 1 F (which is a polynomial functor of degree 1) on vector spaces V with dim( V ) ≥ 6. To convert this theorem into a functorial homotopy-theoretic description of F ( V ), one needs to know in addition what T 1 F ( V ) is when V = 0. Here we show that T 1 F (0) is the standard L -theory space associated with the group ℤ/2, except for a deviation in π 0 . The main corollary is a functorial two-stage decomposition of F ( V ) for dim( V ) ≥ 6 which has the L -theory of the group ℤ/2 as one layer, and a form of unreduced homology of ℝ P ( V ) with coefficients in the L -theory of the trivial group as the other layer. Except for dimension shifts, these are also the layers in the traditional Sullivan-Wall-Quinn-Ranicki decomposition of F ( V ). But the dimension shifts are serious and the SWQR decomposition of F ( V ) is not functorial in V . Because of the functoriality, our analysis of F ( V ) remains meaningful and valid when V = ℝ ∞ .