A Marstrand theorem for measures with polytope density

Given and any centrally symmetric convex polytope , define we prove that if a Radon measure μ has the property then s is an integer. For the case Θ is the Euclidean ball, this result was first proved by Marstrand in 1955 for Hausdorff measure in the plane (Marstrand in Proc Lond Math Soc 3(4):257–302, 1954) and later for general Radon measures in (Marstrand in Trans Am Math Soc 205:369–392, 1964).     

Dynamically convex Finsler metrics and <Emphasis Type="Italic">J</Emphasis>-holomorphic embedding of asymptotic cylinders

We explore the relationship between contact forms on defined by Finsler metrics on and the theory developed by H. Hofer, K. Wysocki and E. Zehnder (Hofer etal. Ann. Math. 148, 197–289, 1998; Ann. Math. 157, 125–255, 2003). We show that a Finsler metric on with curvature K ≥ 1 and with all geodesic loops of length > π is dynamically convex and hence it has either two or infinitely many closed geodesics. We also explain how to explicitly construct J-holomorphic embeddings of cylinders asymptotic to Reeb orbits of contact structures arising from Finsler metrics on with K = 1, thus complementing the results obtained in Harris and Wysocki (Trans. Am. Math. Soc., to appear).      

A note on slant submanifolds of nearly trans-Sasakian manifolds

The geometry of slant submanifolds of a nearly trans-Sasakian manifold is studied when the tensor field Q is parallel. It is proved that Q is not parallel on the submanifold unless it is anti-invariant and thus the result of [CABRERIZO, J. L.—CARRIAZO, A.—FERNANDEZ, L. M.—FERNANDEZ, M.: Slant submanifolds in Sasakian manifolds, Glasg. Math. J. 42 (2000), 125–138] and [GUPTA, R. S.—KHURSHEED HAIDER, S. M.—SHARFUDIN, A.: Slant submanifolds of a trans-Sasakian manifold, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 47 (2004), 45–57] are generalized.     

A Fatou–Bieberbach domain in \mathbb {C}^2 which is not Runge

Since a paper by Rosay and Rudin (Trans. Am. Math. Soc. 310, 47–86, 1988) there has been an open question whether all Fatou–Bieberbach domains are Runge. We give an example of a Fatou–Bieberbach domain Ω in which is not Runge. The domain Ω provides (yet) a negative answer to a problem of Bremermann. Supported by Schweizerische Nationalfonds grant 200021-116165/1.     

Contact totally umbilical submanifolds of a sasakian space form

Summary We define a notion of contact totally umbilical submanifolds of Sasakian space forms corresponds to those of totally umbilical submanifolds of complex space forms. We study a contact totally umbilical submanifold M of a Sasakian space form (c ≠ −3) and prove that M is an invariant submanifold or an anti-invariant submanifold. Furthermore we study a submanifold M with parallel second fundamental form of a Sasakian space form (c ≠ 1) and prove that M is invariant or anti-invariant. Entrata in Redazione il 7 settembre 1976.     

A local and global splitting result for real Kähler Euclidean submanifolds

We show that if a real Kähler Euclidean submanifold is as far as possible of being minimal, then it should split locally as a product of hypersurfaces almost everywhere, possibly in lower codimension. In addition, if the manifold is complete, simply connected and has constant nullity, it should split globally as a product of surfaces in and an Euclidean factor. Several applications are also given.Received: 28 May 2004     

Hyers–Ulam–Rassias Stability of a Quadratic and Additive Functional Equation in Quasi-Banach Spaces

In this paper we establish the general solution of the functional equation

Equivariant gluing constructions of contact stationary Legendrian submanifolds in $${\mathbb {S}^{2n+1}}$$

A contact-stationary Legendrian submanifold of is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact-stationary Legendrian submanifold (actually minimal Legendrian) is the real, equatorial n-sphere S ₀. This paper develops a method for constructing contact-stationary (but not minimal) Legendrian submanifolds of by gluing together configurations of sufficiently many many U(n + 1)-rotated copies of S ₀. Two examples of the construction, corresponding to finite cyclic subgroups of U(n + 1) are given. The resulting submanifolds are very symmetric; are geometrically akin to a ‘necklace’ of copies of S ₀ attached to each other by narrow necks and winding a large number of times around before closing up on themselves; and are topologically equivalent to .     

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians, vol.1, 2 (Zürich, 1994), pp. 526–536. Birkhäuser, Basel, 1995): given a 3-manifold M with boundary ?M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π₁(Σ = ?M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations ${\mathcal{M}_{g,l}:=Hom_\mathcal{C}(\pi_{g,l}, U)/U}The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the international congress of mathematicians, vol.1, 2 (Zürich, 1994), pp. 526–536. Birkh?user, Basel, 1995): given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π₁(Σ = ∂M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations of the fundamental group π _g,l of a punctured Riemann surface Σ _g,l into an arbitrary compact connected Lie group U. This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involution defined on . We show that the involution is induced by a form-reversing involution β defined on the quasi-Hamiltonian space . The fact that has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in Schaffhauser (A real convexity theorem for quasi-Hamiltonian actions, submitted, 25 p, 2007. ). The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained. Supported by the Japanese Society for Promotion of Science (JSPS).     

On codimensions k immersions of m-manifolds for k = 1 and k = m − 2

Let us consider M a closed smooth connected m-manifold, N a smooth (2m − 2)-manifold and a continuous map, with . We prove that if is injective, then f is homotopic to an immersion. Also we give conditions to a map between manifolds of codimension one to be homotopic to an immersion. This work complements some results of Biasi et al. (Manu. Math. 104, 97–110, 2001; Koschorke in The singularity method and immersions of m-manifolds into manifolds of dimensions 2m − 2, 2m − 3 and 2m − 4. Lecture Notes in Mathematics, vol. 1350. Springer, Heidelberg, 1988; Li and Li in Math. Proc. Camb. Phil. Soc. 112, 281–285, 1992).     

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

Approximate Isomorphisms in <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

This paper is a survey on the Hyers–Ulam–Rassias stability of the following Cauchy–Jensen functional equation in C ^*-algebras:

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].      

An extension of Barta’s Theorem and geometric applications

We prove an extension of a theorem of Barta and we give some geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We show that the spectrum of the Nadirashvili bounded minimal surfaces in have positive lower bounds. We prove a stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove generalization of a result of Kazdan–Kramer about existence of solutions of certain quasi-linear elliptic equations. Bessa and Montenegro were partially supported by CNPq Grant.     

Additive group actions on Danielewski varieties and the cancellation problem

Given complex algebraic varieties X and Y of the same dimension, the Cancellation Problem asks if an isomorphism between X  ×  and Y  ×  induces an isomorphism between X and Y. Iitaka and Fujita (J. Fac. Sci. Univ. 24:123–127, 1977) established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski constructed a counterexample using smooth rational affine surfaces. His construction was further generalized by Fieseler (Comment. Math. Helvetici 69:5–27, 1994) and Wilkens (C.R. Acad. Sci. Paris Sér. I Math. 326(9):1111–1116, 1998) to describe a larger class of affine surfaces. Here we introduce higher-dimensional analogues of these surfaces. By studying algebraic actions of the additive group on certain of these varieties, we obtain new counterexamples to the Cancellation Problem in every dimension d ≥ 2.     

Submanifolds of positive Ricci curvature in a Euclidean space

In this paper we study the role of constant vector fields on a Euclidean space R ^n+p in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R ^n+p with mean curvature vector field H and a constant vector field on R ^n+p , the smooth function is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).      

On Hardy’s uncertainty principle for connected nilpotent Lie groups

In (Kaniuth and Kumar in Math. Proc. Camb. Phil. Soc. 131, 487–494, 2001) Hardy’s uncertainty principle for was generalized to connected and simply connected nilpotent Lie groups. In this paper, we extend it further to connected nilpotent Lie groups with non-compact centre. Concerning the converse, we show that Hardy’s theorem fails for a connected nilpotent Lie group G which admits a square integrable irreducible representation and that this condition is necessary if the simply connected covering group of G satisfies the flat orbit condition.     

Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of Sⁿ by looking at the conormal bundle of appropriate submanifolds of Sⁿ. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey–Lawson for submanifolds in Rⁿ in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G₂ and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy. Mathematics Subject Classification (2000): 53-XX, 58-XX.     

Functorial properties of the microsupport and regularity for ind-sheaves

The notion of microsupport and regularity for ind-sheaves was introduced by Kashiwara and Schapira in (Astérisque Soc. Math. France 284:143–164, 2003). In this paper we study the behaviour of the microsupport under several functorial operations and characterize “microlocally” the ind-sheaves that are regular along involutive manifolds. As an application we prove that if a coherent -module is regular along an involutive manifold V (in the sense of (Kashiwara and Oshima Ann. Math. 106:145–200, 1977)), then the ind-sheaf of temperate holomorphic solutions of is regular along V. Another application is the notion of microsupport for sheaves on the subanalytic site, which can be given directly, without using the more complicated language of ind-sheaves.