Locally symmetric minimal affine Lagrangian surfaces in C2

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry.     

Affine geometry of special real submanifolds of Cn

In this paper we propose an affine analogue and generalization of the geometry of special Lagrangian submanifolds of Cⁿ.      

New construction of special Lagrangian submanifolds in <Emphasis Type=Italic>T</Emphasis>*R<Superscript><Emphasis Type=Italic>n</Emphasis></Superscript> equipped with the standard metric

A class of twisted special Lagrangian submanifolds in T＊R^n and a kind of austere submanifold from conormal bundle of minimal surface of R^3 are constructed.     

A Second Main Theorem of Nevanlinna Theory for Meromorphic Functions on Complex Submanifolds in C<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show a second main theorem of Nevalinna theory for meromorphic functions on complex submanifolds in C ⁿ . This has a similar form to the classical one and has a remainder term including Ricci curvature. We also give a concrete computation of the remainder term in the case of nonsingular algebraic submanifolds. Partially supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science.     

Characterization of order 3 algebraic immersed minimal surfaces of <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

In this paper we prove that if is a minimal immersion of a compact surface and , for some homogeneous polynomial f of degree 3 on R ⁴, then, M is a torus and is one of the examples given by Lawson (1970, Complete minimal surfaces in S ³. Ann. Math. 92(2), 335–374).      

On second order minimal Lagrangian sub manifolds in complex space forms

In this paper, we determine all second order minimal Lagrangian submanifolds in complex space forms whose cubic forms have the largest non-trivial continuous symmetries. We describe these minimal Lagrangian submanifolds from the viewpoint of Bryant and study their geometric properties.     

On Hamiltonian-Minimal Lagrangian Tori in $$\mathbb{C}P^2 $$

We obtain some equations for Hamiltonian-minimal Lagrangian surfaces in CP ² and give their particular solutions in the case of tori.     

Two-dimensional quotients of C<Superscript>n</Superscript> are isomorphic to C<Superscript>2</Superscript>/Γ

We will prove that any two-dimensional quotient of an affine space modulo a reductive algebraic group is isomorphic to a quotient of C² modulo a finite group. The proof uses some new results due to Koras and Russell on contractible surfaces with at most quotient singularities and also several results about reductive group actions on affine varieties.     

Free affine actions of unipotent groups on C<Superscript>n</Superscript>

We consider free affine actions of unipotent complex algebraic groups on Cⁿ and prove that such actions admit an analytic geometric quotient if their degree is at most 2. Moreover, we classify free affine C²-actions on Cⁿ of degree n - 1 and n - 2. For every n > 4, an action of degree n - 2 appears in the classification whose quotient topology is not Hausdorff.     

Holomprhic vector bundles on C<Superscript>2</Superscript>∖{0}

Here we show the existence of a rank 2 holomorphic vector bundleE on C²∖{0} without any holomorphic rank 1 subsheaf. Hence, contrary to the algebraic case, there are open subsets of dimension 2 Stein manifolds with holomorphic vector bundles which are not filtrable in any weak sense.     

Tensor Products of Hilbert Modules Over Locally <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

In this paper the tensor products of Hilbert modules over locally C ^*-algebras are defined and their properties are studied. Thus we show that most of the basic properties of the tensor products of Hilbert C ^*-modules are also valid in the context of Hilbert modules over locally C ^*-algebras.     

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

Surfaces with Vanishing Moebius Form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

An important Moebius invariant in the theory of Moebius surfaces in S^n is the so-called Moebius form. In this paper,we give a complete classification of surfaces in S^n with vanishing Moebius form under the Moebius transformation group.     

Hyperbolic linear Weingarten surfaces in ℝ<Superscript>3</Superscript>

A hyperbolic linear Weingarten surface in ℝ³ is a surface M whose mean and Gaussian curvatures satisfy the relationship 2aH +bK = c for real numbers a, b, c such that a²+bc < 0. In this work we obtain a representation for such a surface in terms of its Gauss map when, more generally, a, b, c are functions on M. We also study the completeness of such surfaces and describe a procedure to construct complete examples from solutions of the sine-Gordon equation. The first author is partially supported by Junta de Comunidades de Castilla-La Mancha, Grant no. PAI-05-034. The first and second authors are partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746.     

Stable nonperiodic points of two-dimensional C<Superscript>1</Superscript>-diffeomorphisms

It is proved that there exist two-dimensional diffeomorphisms with countably many stable periodic points in a neighborhood of a homoclinic point. The characteristic exponents of these points are negative and bounded away from zero.     

A criterion for <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript>-rectifiability of sets

The following result is proved: Let be a n-dimensional C¹-submanifold of R^N which is domain of a given _nR^N-valued map of class C¹. Then the set of all points P such that (P) is non-zero, simple and enveloped by T_P is C²-rectifiable. As a corollary we get a criterion for the C²-rectifiability of a rectifiable set based on the rectifiability of some generalized Gauss lift to the Grassmanian bundle R^N×G(N,n). Mathematics Subject Classification (2000) Primary 49Q15, 53A07; Secondary 49Q20, 49N60     

Evolution Equations for Special Lagrangian 3-Folds in C3

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C ^m . The previous paper (Math. Ann. 320 (2001), 757–797), defined the idea of evolution data, which includes an (m – 1)-submanifold P in R ⁿ , and constructed a family of special Lagrangian m-folds N in C ^m , which are swept out by the image of P under a 1-parameter family of affine maps _t : R ⁿ C ^m , satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C³. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C³.Our main results are a number of new families of special Lagrangian 3-foldsin C³, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R³ or ¹× R². Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.     

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: