Annali di Matematica Pura ed Applicata最新文献

In this paper we describe all invariant complex Dirac structures with constant real index on a maximal flag manifold in terms of the roots of the Lie algebra which defines the flag manifold. We also completely classify these structures under the action of B-transformations.

We revisit the well-known Brezis-Nirenberg problem

where (varepsilon >0) and (Omega subset mathbb {R}^N) are a smooth bounded domain with (Nge 3). The existence of multi-bump solutions to above problem for small parameter (varepsilon >0) was obtained by Musso and Pistoia (Indiana Univ Math J 51:541–579, 2002). However, to our knowledge, whether the multi-bump solutions are non-degenerate that is open. Here, we give some straightforward answer on this question under some suitable assumptions for the Green’s function of (-Delta ) in (Omega ), which enriches the qualitative analysis on the solutions of Brezis-Nirenberg problem and can be viewed as a generalization of Grossi (Nonlinear Differ Equ Appl 12:227–241, 2005) where the non-degeneracy of a single-bump solution has been proved. And the main idea is the blow-up analysis based on the local Pohozaev identities.

In 1951, Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in Euclidean space are the round (geometrical) spheres. These results were generalized by S. S. Chern and then by Eschenburg and Tribuzy for surfaces homeomorphic to the sphere in Riemannian manifolds with constant sectional curvature whose mean curvature function satisfies some bound on its differential. In this paper, we extend these results for surfaces in a wide class of warped product manifolds, which includes, besides the classical space forms of constant sectional curvature, the de Sitter–Schwarzschild manifolds and the Reissner–Nordstrom manifolds, which are time slices of solutions of the Einstein field equations of general relativity.

We show that 3-Sasaki structures admit a natural description in terms of projective differential geometry. First we establish that a 3-Sasaki structure may be understood as a projective structure whose tractor connection admits a holonomy reduction, satisfying a particular non-vanishing condition, to the (possibly indefinite) unitary quaternionic group ({text {Sp}}(p,q)). Moreover, we show that, if a holonomy reduction to ({text {Sp}}(p,q)) of the tractor connection of a projective structure does not satisfy this condition, then it decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) 3-Sasaki structures and a closed separating hypersurface at infinity with respect to the 3-Sasaki metrics. It is shown that the latter hypersurface inherits a Biquard–Fefferman conformal structure, which thus (locally) fibers over a quaternionic contact structure, and which in turn compactifies the natural quaternionic Kähler quotients of the 3-Sasaki structures on the open manifolds. As an application, we describe the projective compactification of (suitably) complete, non-compact (indefinite) 3-Sasaki manifolds and recover Biquard’s notion of asymptotically hyperbolic quaternionic Kähler metrics.

Let G be a finite group, (mathbb {F}) be one of the fields (mathbb {Q},mathbb {R}) or (mathbb {C}), and N be a non-trivial normal subgroup of G. Let ({textrm{acd}}^{*}_{{mathbb {F}}}(G)) and ({textrm{acd}}_{{mathbb {F}}, textrm{even}}(G|N)) be the average degree of all non-linear (mathbb {F})-valued irreducible characters of G and of even degree (mathbb {F})-valued irreducible characters of G whose kernels do not contain N, respectively. We assume the average of an empty set is zero for more convenience. In this paper we prove that if (textrm{acd}^*_{mathbb {Q}}(G)< 9/2) or (0<textrm{acd}_{mathbb {Q},textrm{even}}(G|N)<4), then G is solvable. Moreover, setting (mathbb {F} in {mathbb {R},mathbb {C}}), we obtain the solvability of G by assuming ({textrm{acd}}^{*}_{{mathbb {F}}}(G)<29/8) or (0<{textrm{acd}}_{{mathbb {F}}, textrm{even}}(G|N)<7/2), and we conclude the solvability of N when (0<{textrm{acd}}_{{mathbb {F}}, textrm{even}}(G|N)<18/5). Replacing N by G in ({textrm{acd}}_{{mathbb {F}}, textrm{even}}(G|N)) gives us an extended form of a result by Moreto and Nguyen. Examples are given to show that all the bounds are sharp.

An almost abelian Lie group is a solvable Lie group with a codimension one normal abelian subgroup. We characterize almost Hermitian structures on almost abelian Lie groups where the almost complex structure is harmonic with respect to the Hermitian metric. Also, we adapt the Gray–Hervella classification of almost Hermitian structures to the family of almost abelian Lie groups. We provide several examples of harmonic almost complex structures in different Gray–Hervella classes on some associated compact almost abelian solvmanifolds.

For a large class of operators acting between weighted (ell ^infty) spaces, exact formulas are given for their norms and the norms of their restrictions to the cones of nonnegative sequences; nonnegative, nonincreasing sequences; and nonnegative, nondecreasing sequences. The weights involved are arbitrary nonnegative sequences and may differ in the domain and codomain spaces. The results are applied to the Cesàro and Copson operators, giving their norms and their distances to the identity operator on the whole space and on the cones. Simplifications of these formulas are derived in the case of these operators acting on power-weighted (ell ^infty). As an application, best constants are given for inequalities relating the weighted (ell ^infty) norms of the Cesàro and Copson operators both for general weights and for power weights.

We are concerned with a one-dimensional flow of a compressible fluid which may be seen as a simplification of the flow of fluid in a long thin pipe. We assume that the pipe is on one side ended by a spring. The other side of the pipe is let open—there we assume either inflow or outflow boundary conditions. Such situation can be understood as a toy model for human lungs. We tackle the question of uniqueness and existence of a strong solution for a system modeling the above process, special emphasis is laid upon the estimate of the maximal time of existence.

The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to the Böttcher–Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras. A technical point of interest is that the results work over the octonions as well as the associative Hurwitz algebras.

We consider solitary wave solutions of a two-component Novikov system, which is a coupled Camassa-Holm type system with cubic nonlinearity. Inspired by the methods established by Constantin and Strauss in [6, 7], we prove that the smooth solitary waves and non-smooth peakons to the system are both orbitally stable.