The influence of the boundary behavior on isometric immersions in the hyperbolic space

This paper studies how the behavior of a proper isometric immersion into the hyperbolic space is influenced by its behavior at infinity. Our first result states that a proper isometric minimal immersion into the hyperbolic space with the asymptotic boundary contained in a sphere reduces codimension. This result is a corollary of a more general one that establishes a sharp lower bound for the sup-norm of the mean curvature vector of a Proper isometric immersion into the Hyperbolic space whose Asymptotic boundary is contained in a sphere. We also prove that a properly immersed hypersurface with mean curvature satisfying sup _{p∈Σ ||H(p)|| < 1} has no isolated points in its asymptotic boundary. Our main tool is a Tangency principle for isometric immersions of arbitrary codimension. This work is partially supported by CAPES, Brazil.     

The multi-dimensional pencil phenomenon for Laguerre heat-diffusion maximal operators

We investigate in detail the mapping properties of the maximal operator associated with the heat-diffusion semigroup corresponding to expansions with respect to multi-dimensional standard Laguerre functions . Our interest is focused on the situation when at least one coordinate of the type multi-index α is smaller than 0. For such parameters α the Laguerre semigroup does not satisfy the general theory of semigroups, and the behavior of the associated maximal operator on L ^p spaces is found to depend strongly on both α and the dimension. A. Nowak was supported in part by MNiSW Grant N201 054 32/4285.     

Minimal surfaces,Hopf differentials and the Ricci condition

We deal with minimal surfaces in a sphere and investigate certain invariants of geometric significance, the Hopf differentials, which are defined in terms of the complex structure and the higher fundamental forms. We discuss the holomorphicity of Hopf differentials and provide a geometric interpretation for it in terms of the higher curvature ellipses. This motivates the study of a class of minimal surfaces, which we call exceptional. We show that exceptional minimal surfaces are related to Lawson’s conjecture regarding the Ricci condition. Indeed, we prove that, under certain conditions, compact minimal surfaces in spheres which satisfy the Ricci condition are exceptional. Thus, under these conditions, the proof of Lawson’s conjecture is reduced to its confirmation for exceptional minimal surfaces. In fact, we provide an affirmative answer to Lawson’s conjecture for exceptional minimal surfaces in odd dimensional spheres or in S ^4m .     

B.Y. Chen inequalities for slant submanifolds in Sasakian space forms

In the present paper, we establish Chen inequalities for slant submanifolds in Sasakian space forms, by using subspaces orthogonal to the Reeb vector field ξ.     

Embedded Rotational Hypersurfaces with Constant Scalar Curvature in Sn: A Correction to a Statement in M. L. Leite, Manuscripta Math. 67 (1990), 285-304.

We prove that there exist (n − 1)-dimensional compact embedded rotational hypersurfaces with constant scalar curvature (n − 1)(n − 2)S (S > 1) of S ⁿ other than product of spheres for 4 ≤ n ≤ 6. As a result, we prove that Leite’s Assertion was incorrect.The project is supported by the grant No. 10531090 of NSFC.     

Morse indices and the number of maximum points of some solutions to a two-dimensional elliptic problem

In this note, we consider the problem

Rearrangements and radial graphs of constant mean curvature in hyperbolic space

We investigate the problem of finding smooth hypersurfaces of constant mean curvature in hyperbolic space, which can be represented as radial graphs over a subdomain of the upper hemisphere. Our approach is variational and our main results are proved via rearrangement techniques. The second author was partially supported by NSF grant DMS 0603707.     

A parabolic almost monotonicity formula

We prove the parabolic counterpart of the almost monotonicity formula of Caffarelli, Jerison and Kening for pairs of functions u _±(x, s) in the strip satisfying

Geodesics of Sasakian Metrics on Tensor Bundles

On the basis of the phase completion the notion of vertical and horizontal lifts of vector fields is defined in the tensor bundles over a Riemannian manifold. Such a tensor bundle is made into a manifold with a Riemannian structure of special type by endowing it with Sasakian metric. The components of the Levi-Civita and other metric connections with respect to Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done, it is shown that it is possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifolds. Dedicated to the memory of Vladimir Vishnevskii (1929-2007)     

Symmetric Generalized Galois Logics

Symmetric generalized Galois logics (i.e., symmetric gGls) are distributive gGls that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGls by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGls. We generalize the weak distributivity laws between fusion and fission to interactions of certain monotone operations within distributive super gGls. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGls and classes of structures for them.      

A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two

We prove a new regularity result for systems of nonlinear elliptic equations with quadratic Jacobian type nonlinearity in dimension two. Our proof is based on an adaptation of John Lewis’ method which has not been used for such systems so far. Parts of this work have been done while the second and the third author had been enjoying the hospitality of the Department of Mathematics of the University of Pittsburgh. P.H. was supported by NSF grant DMS-0500966. P.S. was partially supported by the MNiSzW grant no 1 PO 3A 005 29. X.Z. was supported by the Academy of Finland, project 207288.     

Harmonic 1-Forms on Compact f-Manifolds

We consider a Riemannian manifold with a compatible f-structure which admits a parallelizable kernel. With some additional integrability conditions it is called -manifold. This class of manifolds is a natural generalization of the Sasakian manifolds. We study properties of harmonic 1-forms on such a manifold and deduce some topological properties. Research supported by the Italian MIUR 60% and GNSAGA. Professor J.J. Konderak died just during the preparation of this paper. His contribution has been substantial. Note of the Editor-in-Chief. Professor Jerzy J. Konderak passed away on September 14, 2005, at the age of 49. Born on February 12, 1956, in Krakow, Poland, he received his Ph.D. in 1986 from the Jagiellonian University of Krakow. Since 2003 he was associated professor of Mathematics at the Faculty of Sciences of the University of Bari. The main area of his scientific interests was Differential Geometry. Professor Konderak was a very gifted researcher and was appreciated by students and colleagues for his passion and devotion to Mathematics and its teaching. All the colleagues of the Department of Mathematics of the University of Bari will remember him not only as a clever mathematician but also as a men of rare quality.     

An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $${\mathbb H^2 \times \mathbb R}$$

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in . As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary Γ_∞ is a Jordan curve homologous to zero in such that Γ_∞ is contained in a slab between two horizontal circles of with width equal to π. We construct vertical minimal graphs in over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains Ω in are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition. The first author wish to thank Laboratoire Géométrie et Dynamique de l’Institut de Mathématiques de Jussieu for the kind hospitality and support. The authors would like to thank CNPq, PRONEX of Brazil and Accord Brasil-France, for partial financial support.     

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.     

Low regularity classes and entropy numbers

We note a sharp embedding of the Besov space into exponential classes and prove entropy estimates for the compact embedding of subclasses with logarithmic smoothness, considered by Kashin and Temlyakov. A.S. was supported in part by the National Science Foundation grant 0652890.     

Hamiltonian systems of PDEs with selfdual boundary conditions

Selfdual variational calculus is developed further and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, and Hamiltonian systems of PDEs. This allows for the resolution of such equations under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and anti-periodic orbits, but also yield new ones such as “periodic orbits up to an isometry” for evolution equations that may not have periodic solutions. In the process, we introduce a method for perturbing selfdual functionals in order to induce coercivity and compactness, without destroying the selfdual character of the system. N. Ghoussoub was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. A. Moameni’s research was supported by a postdoctoral fellowship at the University of British Columbia.     

Hardy and BMO spaces associated to divergence form elliptic operators

Consider a second order divergence form elliptic operator L with complex bounded measurable coefficients. In general, operators based on L, such as the Riesz transform or square function, may lie beyond the scope of the Calderón–Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some L ^p spaces. In this work we develop a theory of Hardy and BMO spaces associated to L, which includes, in particular, a molecular decomposition, maximal and square function characterizations, duality of Hardy and BMO spaces, and a John–Nirenberg inequality. S. Hofmann was supported by the National Science Foundation.     

Triply periodic minimal surfaces bounded by vertical symmetry planes

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz–Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces. S. Fujimori was partially supported by JSPS Grant-in-Aid for Young Scientists (Start-up) 19840035. M. Weber’s material is based upon work for the NSF under Award No. DMS-0139476.     

Bending the helicoid

We construct Colding–Minicozzi limit minimal laminations in open domains in \({\mathbb{R}}^3\) with the singular set of C ¹-convergence being any properly embedded C ^1,1-curve. By Meeks’ C ^1,1-regularity theorem, the singular set of convergence of a Colding–Minicozzi limit minimal lamination \({\mathcal{L}}\) is a locally finite collection \(S({\mathcal{L}})\) of C ^1,1-curves that are orthogonal to the leaves of the lamination. Thus, our existence theorem gives a complete answer as to which curves appear as the singular set of a Colding–Minicozzi limit minimal lamination. In the case the curve is the unit circle \({\mathbb{S}}^1(1)\) in the (x ₁, x ₂)-plane, the classical Björling theorem produces an infinite sequence of complete minimal annuli H _n of finite total curvature which contain the circle. The complete minimal surfaces H _n contain embedded compact minimal annuli \(\overline{H}_n\) in closed compact neighborhoods N _n of the circle that converge as \(n \to \infty\) to \(\mathbb {R}^3 - x_3\) -axis. In this case, we prove that the \(\overline{H}_n\) converge on compact sets to the foliation of \(\mathbb {R}^3 - x_3\) -axis by vertical half planes with boundary the x ₃-axis and with \({\mathbb{S}}^1(1)\) as the singular set of C ¹-convergence. The \(\overline{H}_n\) have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.     

The initial boundary value problem for the nonlinear parabolic equation in unbounded domain

In this paper we consider the mixed problem for the equation u _tt  + A ₁ u + A ₂(u _t ) + g(u _t ) = f(x, t) in unbounded domain, where A ₁ is a linear elliptic operator of the fourth order and A ₂ is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially, in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution at infinity.