Showing papers in "Kodai Mathematical Journal in 1991"

Null finite type hypersurfaces in space forms

Abstract: In [5], Chen gives a classification of null 2-type surfaces in the Euclidean 3-space and he shows in [6] that a similar characterization cannot be given for a surface in the Euclidean 4-space. In fact, helical cylinders in Euclidean 4-space are characterized as those surfaces of null 2-type and constant mean curvature. In this paper we give a characterization of null 2-type hypersurfaces in a space of constant sectional curvature Mn+1(k) and an approach to hypersurfaces of null 3-type. Indeed, we get a generalization of Chen’s paper [5] not only by considering hypersurfaces, but also taking them in space forms. In spherical and hyperbolic cases we show that there is no null 2-type hypersurface, so that the Euclidean case becomes the most attractive situation where our classification works on. Actually, we show that Euclidean hypersurfaces of null 2-type and having at most two distinct principal curvatures are locally isometric to a generalized cylinder. Why the hypothesis on principal curvatures? First, we think this is the most natural one, because, after Chen’s paper, we know that cylinders are the only surfaces of null 2-type in Euclidean 3-space. Secondly, it is well-known that a Euclidean isoparametric hypersurface has at most two distinct principal curvatures, so that if it has exactly two, then one of them has to be zero. Our classification depends strongly on that isoparametricity condition. Finally, bounding the number of principal curvatures is not as restrictive as one could hope. As a matter of fact, the families of conformally flat and rotational hypersurfaces satisfy that hypothesis and both are sufficiently large so that it is worth trying to give a characterization of some subfamily of them in order to get along in their classifications. To this effect, we characterize rotational and conformally flat hypersurfaces of null 2-type. As for hypersurfaces of null 3-type one immediately sees that they are not difficult to handle when they have constant mean curvature, because a nice formula for ∆2H can be given. In that case, we show that there is no spherical or hyperbolic hypersurface of null 3-type. It turns out again that our only hope to get some more information concerns with Euclidean hypersurfaces. Now, following a similar reasoning as in the null 2-type case, we are able to say that there is no Euclidean hypersurface of null 3-type having constant mean curvature and at most two distinct principal curvatures. We wish to thank to Prof. M. Barros for many valuable comments and suggestions.

Linearly independent, orthogonal and equivariant immersions

Abstract: In this article we define the notions of linearly independent and orthogonal immersions and introduce the notion of adjoint hyperquadrics of linearly independent immersions. We investigate the relations between linearly indedendent immersions, orthogonal immersions, equivariant immersions and adjoint hyperquadrics. Several results in this respect are obtained.

Nonlinear Volterra integrodifferential evolution inclusions and optimal control

Abstract: In this paper we examine integrodifferential evolution inclusions of the Volterra type driven by time dependent, monotone, hemicontinuous operators. We prove two existence theorems; one for convex valued perturbations and the other for nonconvex valued ones. We also establish a topological property of the solution set of the "convex" problem. Then we prove a result on the continuous dependence of the solutions on the data of the problem (sensitivity analysis). We also consider a random version of the inclusion and prove that it admits a random solution. Then we pass to optimal control problems. First we establish the existence of optimal admissible pairs and then using the notions of epigraphical and G-convergences, we obtain a variational stability result. Finally we work in detail two parabolic distributed parameter optimal control problems, illustrating the applicability of our work. List of symbols Ω: Upper case Greek letter omega Wr: Σ: Upper case Greek letter sigma &: S: S subscript q dH: SPF: S superscript p, subscript F φ : ω: Lower case Greek letter omega φ: Wpq: W subscript pq δ: L: L superscript q BM: L : L superscript p a : τ: Lower case Greek letter tau β : ε : Lower case Greek letter epsilon γ: η : Lower case Greek letter eta Γ: X\ Script L aΐj . W subscript r Script P d subscript H Lower case Greek letter phi Lower case Greek letter psi Lower case Greek letter delta B subscript M Lower case Greek letter alpha Lower case Greek letter beta Lower case Greek letter gamma Upper case Greek letter gamma a superscript n, subscript ij

Value distribution of meromorphic mappings into compactified locally symmetric spaces

Abstract: There have been many works to generalize the Nevanlinna theory (especially, his second main theorem) to higher dimensional case (cf., e.g., [5], [17], [21]). While the so called equidimensional holomorphic or meromorphic mappings / : W—>V between algebraic varieties have been well studied, we do not know very much about f:W-+V with άimW<άimV (cf. Noguchi [13], [14] and Siu [20]). So far, we have to put some special restriction on target spaces or on the divisors of the target spaces. In this paper, we will establish an inequality of the second main theorem type for meromorphic mappings into a compactification of a locally symmetric space. Let 3) be a bounded symmetric domain in C m and Γ a neat arithmetic discrete subgroup of the holomorphic transformation group Aut (3)) of 3). Let / : C->Γ\\β)

A unicity theorem for meromorphic mappings into compactified locally symmetric spaces

Abstract: The classical theorem of Nevanlinna states that non-constant holomorphic mappings /, g: C-»Pi(C) satisfying f~\\ai)=g~\\at) with multiplicities for distinct five points au ••• a^P^C) are identical ([11]). The unicity theorems of this type for holomorphic (or meromorphic) mappings were studied by several authors (cf., e.g., [4], [5], [6] and [14]). For instance, in [6], H. Fujimoto studied meromorphic mappings / : Cn-*Pm(C), using BoreΓs theorem and obtained many interesting results. On the other hand, S. Drouilhet [5] proved a unicity theorem of another type for meromorphic mappings / : M—>V, where M is a smooth affine variety and V is a smooth projective variety with dim V <£ dim M. He used the second main theorem for meromorphic mappings due to Shiffman [15]. In this paper, we prove some unicity theorems for meromorphic mappings of a finite analytic covering space over C into a smooth toroidal compactification of a locally symmetric space, by making use of a second main theorem proved in [1], Let 3) be a bounded symmetric domain in C and ΓcAut (3)) a neat arithmetic group. Let y be a positive rational number such that the holomorphic sectional curvature of the Bergman metric on 3) is bounded by — γ from above. We denote by Γ\\3) a smooth toroidal compactification of Γ\\3) such that D— Γ\\3)—Γ\\3) is a hypersurface with only normal crossings. Let c: Γ\\3)->PN(C) be a non-constant holomorphic mapping and \\_H~]->PN{C) the hyperplane bundle over PN(C). Let π: X-^C n be a finite analytic covering with ramification divisor R. Then we have the following unicity theorem for meromorphic mappings / : X->Γ\\3) in the case l

Generalized Hopf manifolds, locally conformal Kaehler structures and real hypersurfaces

Abstract: We study the geometry of submanifolds of complex Hopf manifolds endowed with the (locally conformal Kaehler) Boothby metric. 1. Generalized Hopf manifolds and the Boothby metric. Let fleC, 0 l , generated by z^az, z(=W. Then Ga acts freely and properly discontinuously on W, see [28], vol. II, p. 137, so that the quotient space Hna=W/Ga becomes in a natural way a complex w-dimensional manifold. This is the well known complex Hopf manifold. In their attempt to construct complex structures on products SxL, where S is the unit circle and L an odd dimensional homotopy sphere, E. Brieskorn & A. Van de Ven, [3], have generalized Hopf manifolds a follows. Let n > l and (fe0, ••• , bn)^Z , bj^l, 0£j£n. Let (z0, ••• , zn) be the natural complex coordinates on C n + 1 . Define X(b) = X2n(b0, ••• , fc»)cC n+1 by the equation: Then X(b) ia an aίϊine algebraic variety with one singular point at the origin of C if bj^2, / = 0 , -" n (and without singularities if b3—\ for at least one /). Next B{b)—X(b)— {0} is a complex n-dimensional manifold, referred hereafter as the Brieskorn manifold determined by the integers b0, ••• , bn. See [2]. There is a natural holomorphic action of C on B(b) given by: t(Zo, —, ^ n ) = ( ^ O e X p ( y ^ ) , ••• , Zn ΘXp (—-y^ ) ) (1) where f e C , wa=— log \a \— iΦa, Φα=arctan(/m(α)//?β(fl)), —π/2 D x 5»(« defined by f(a, jc)=(α, ί/αx), for any a e D 1 , xe=B(b), where £/αe=GL(n+l, C) is the matrix: Note that / is an automorphism of DxBφ). The action of GL(n+l, C) on C n + 1 induces an action of Z « { / 7 m e Z } on DxBφ). Let: be the quotient space. We establish the following: THEOREM 1. X is a complex n-dimensional manifold. Moreover, if n—2, then there exists a surjective holomorphic map π: X-^D which makes X into a complex analytic family of compact complex surfaces, for any a^D there is a diffeomorphism between π~\a) and Hl(b). We recall that a triple (X, π, M) is a complex analytic family of compact complex manifolds if X, M are complex manifolds and π: X-*M is a proper holomorphic map which is of maximal rank at all points of X. Then each fibre π~\a\ G G M , is a compact complex manifold. Note that the action (1) of Z on Bφ) generalizes slightly the one in [3], p. 390. There B(l, ••• , 1)/Z is diffeomorphic to W/G1/e. The proof of Theorem 1 is organized in several steps, as follows. STEP 1. Z acts freely on DxB(b). Let (a, x) be a fixed point of / m , m e Z . Thus Ufx = x, and consequently: for O^tj ^n. Since at least one z3 is non-zero, it follows that m=0. STEP 2. {//meZ} is a properly discontinuous group of analytic transformations of DxBψ). Let KdD, LcB(b) be compact subsets. It is enough to show that the set of all m e Z with the property:

The second variation of the dirichlet energy on contact manifolds

Abstract: REFERENCES[ 1 ] D. E. BLAIR, Contact Manifolds in Riemannian Geometry, Lect. Notes in Math.Vol. 509, Springer, Berlin.[ 2 ] D. E. BLAIR, Critical associated metrics on contact manifolds, J. Aust. Math. Soc.37 (1984), 82-88.[ 3 ] D. E. BLAIR, Critical associated metrics on contact manifolds III. preprint.[4] D.E. BLAIR, On the set of metrics associated to a sympletic or contact form,Bull. Inst. Math. Acad. Sinica. 11 (1983), 297-308.[ 5 ] S. S. CHERN AND R. S. HAMILTON, On Riemannian metrics adapted to three-dimensional contact manifolds, Lect. Notes in Math. Vol. 1111, Springer, Berlin,279-305.[6] D. EBIN, The manifold of Riemannian metrics, Proc. Symp. Pure Math. AMS. 15(1970), 11-40.[ 7 ] S. SASAKI, Almost Contact Manifolds, Lecture Notes. Tόhoku Univ. Vol. 1 1965,Vol. 2 1967, Vol. 3 1968.[ 8 ] S. TANNO, Variational problems on contact Riemannian manifolds, Trans. Amer.Math. Soc. 314 (1989), 349-379.[ 9 ] K. YANO AND M. KON, Structures on Manifolds, World Scientific. 1984.

Inner radii of Teichmüller spaces of finitely generated Fuchsian groups

Abstract: Let Γ be a Fuchsian group keeping the lower half plane L invariant. The Teichmuller space T(Γ) of Γ is a bounded domain of the Banach space B(L,Γ) of bounded quadratic differentials for Γ. The inner radius i(Γ) of T{Γ) is the radius of the maximal ball in B(L, Γ) centered at the origin which is included in T(Γ). If T(Γ) is not a single point, then by a theorem of Ahlfors-Weill [3] it holds that i{Γ)^2. In particular, if Γ is finitely generated of the first kind and if T(Γ) is not a single point, then the strict inequality i(Γ)>2 holds (cf. [10]). Denote by I(Γ) inf iiWΓW'), where the infimum is taken over for all quasiconformal automorphisms W of the upper half plane compatible with Γ. Recently T. Nakanishi [10] proved the following.