Showing papers in "Tohoku Mathematical Journal in 2008"
TL;DR: In this article, the concept of intrinsic ultracontractivity was extended to non-symmetric semigroups and proved the intrinsic ultra-condractivity of non- symmetric second order elliptic operators in bounded Lipschitz domains.
Abstract: We extend the concept of intrinsic ultracontractivity to non-symmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of non- symmetric second order elliptic operators in bounded Lipschitz domains.
51 citations
TL;DR: In this paper, the existence of minimal conformal immersions in the product space was established for a general class of surfaces, where the first factor is a Riemannian surface and the second factor is the set of real numbers.
Abstract: We establish the definition of associate and conjugate conformal minimal isometric immersions into the product spaces, where the first factor is it Riemannian surface and the other is the set of real numbers. When the Gaussian Curvature of the first factor is nonpositive. we prove that an associate surface of it minimal vertical graph over a convex domain is still a vertical graph. This generalizes a well-known result due to R. Krust. Focusing the case when the first factor is the hyperbolic plane, it is known that in certain class of surfaces, two minimal isometric immersions are associate. We show that this is not true in general. In the product ambient space, when the first factor is either the hyperbolic plane or the two-sphere, we prove that the conformal metric and the Hopf quadratic differential determine it simply connected minimal conformal immersion, up to an isometry of the ambient space. For these two product spaces, we derive the existence of the minimal associate family.
44 citations
TL;DR: In this paper, the global behavior of weakly stable constant mean curvature hypersurfaces in a Riemannian manifold was studied by using harmonic function theory, and it was shown that any complete non-compact hypersurface in Euclidean space must have only one end.
Abstract: We study the global behavior of weakly stable constant mean curvature
hypersurfaces in a Riemannian manifold by using harmonic function theory. In
particular, a complete oriented weakly stable minimal hypersurface in the
Euclidean space must have only one end. Any complete noncompact weakly stable
hypersurface with constant mean curvature $H$ in the 4 and 5 dimensional
hyperbolic spaces has only one end under some restrictions on $H$.
34 citations
TL;DR: In this article, the authors study sequences orthonormal with respect to an associated bilinear form and measure properties of such sequences by a positive, possibly unbounded angle operator which is formally orthogonal as a matrix.
Abstract: In a separable complex Hilbert space endowed with an isometric conjugate-linear involution, we study sequences orthonormal with respect to an associated bilinear form. Properties of such sequences are measured by a positive, possibly unbounded angle operator which is formally orthogonal as a matrix. Although developed in an abstract setting, this framework is relevant to a variety of eigenvector interpolation problems arising in function theory and in the study of differential operators.
25 citations
TL;DR: In this article, a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack is given in terms of a collection of line bundles and sections which satisfy certain conditions.
Abstract: We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.
22 citations
TL;DR: In this paper, the authors derived the exact dead-core rates under a dynamical theory assumption and constructed some special solutions for the corresponding Cauchy problem satisfying this dynamical theories.
Abstract: We study the solution of the heat equation with a strong absorption. It is
well-known that the solution develops a dead-core in finite time for a large
class of initial data. It is also known that the exact dead-core rate is faster
than the corresponding self-similar rate. By using the idea of matching, we
formally derive the exact dead-core rates under a dynamical theory assumption.
Moreover, we also construct some special solutions for the corresponding Cauchy
problem satisfying this dynamical theory assumption. These solutions provide
some examples with certain given polynomial rates.
20 citations
TL;DR: In this article, the authors classify numerical Campedelli surfaces with an involution, i.e., an automorphism of order 2, and show that the bicanonical map of $S$ is composed with the involution if and only if it has six isolated fixed points.
Abstract: Numerical Campedelli surfaces are minimal surfaces of general type with vanishing
geometric genus and canonical divisor with self-intersection 2. Although they
have been studied by several authors,their complete classification is not known. In this paper we classify numerical Campedelli surfaces with an involution,
i.e., an automorphism of order 2. First we show that an involution on a
numerical Campedelli surface $S$ has either four or six isolated fixed points,
and the bicanonical map of $S$ is composed with the involution if and only if
the involution has six isolated fixed points. Then we study in detail each of
the possible cases, describing also several examples.
18 citations
TL;DR: In this paper, the authors studied the blow-up profile of radial solutions of a semilinear heat equation with an exponential source term and showed that solutions which can be continued beyond blowup possess a nonconstant self-similar blowup profile.
Abstract: We study the blow-up profile of radial solutions of a semilinear heat equation with an exponential source term. Our main aim is to show that solutions which can be continued beyond blow-up possess a nonconstant selfsimilar blow-up profile. For some particular solutions we determine this profile precisely.
11 citations
TL;DR: In this paper, a class of nonlinear differential equations of second Painleve type with a single exception admit the quasi-Painleve property along a rectifiable curve, that is, for general solutions, every movable singularity defined by a Rectifiable curve is at most an algebraic branch point.
Abstract: We present a class of nonlinear differential equations of second Painleve type. These equations, with a single exception, admit the quasi-Painleve property along a rectifiable curve, that is, for general solutions, every movable singularity defined by a rectifiable curve is at most an algebraic branch point. Moreover we discuss the global many-valuedness of their solutions. For the exceptional equation, by the method of successive approximation, we construct a general solution having a movable logarithmic branch point.
11 citations
TL;DR: In this paper, the authors studied the stability of direct images of vector bundles by Frobenius morphisms up to torsions and proved that the direct images are semistable with respect to the canonical line bundle.
Abstract: We study the stability of direct images by Frobenius morphisms. First, we compute the first Chern classes of direct images of vector bundles by Frobenius morphisms modulo rational equivalence up to torsions. Next, introducing the canonical filtrations, we prove that if $X$ is a nonsingular projective minimal surface of general type with semistable $\Omega_X^1$ with respect to the canonical line bundle $K_X$, then the direct images of line bundles on $X$ by Frobenius morphisms are semistable with respect to $K_X$.
9 citations
TL;DR: In this article, the authors define a new class of manifolds called $n$-Sasakian manifolds that enjoy remarkable geometric properties and make links to the study of isoparametric hypersurfaces.
Abstract: We define a new class of manifolds called $n$-Sasakian manifolds that enjoy remarkable geometric properties. We furnish examples of such manifolds and make links to the study of isoparametric hypersurfaces. We demonstrate that these examples carry Einstein metrics.
TL;DR: In this paper, it was shown that there are three inflection points on a given anti-convex curve such that the tangent lines at these three points cross the curve only once.
Abstract: A simple closed curve in the real projective plane is called anti-convex if for each point on the curve, there exists a line which is transversal to the curve and meets the curve only at that given point. Our main purpose is to prove an identity for anti-convex curves that relates the number of independent (true) inflection points and the number of independent double tangents on the curve. This formula is a refinement of the classical Mobius theorem. We also show that there are three inflection points on a given anti-convex curve such that the tangent lines at these three inflection points cross the curve only once. Our approach is axiomatic and can be applied in other situations. For example, we prove similar results for curves of constant width as a corollary.
TL;DR: In this paper, the authors define on a manifold $X$ a wedge product $S \wedge T$ of a closed positive, smooth outside a proper analytic subset $Y$ of $X$, and a positive pluriharmonic $(k,k)$-current $T$ when $k$ is less than the
Abstract: We define on a manifold $X$ a wedge product $S \wedge T$ of a closed positive
(1,1)-current $S$, smooth outside a proper analytic subset $Y$ of $X$, and a
positive pluriharmonic $(k,k)$-current $T$, when $k$ is less than the
codimension of $Y$ Using this tool, we prove that if $M$ is a compact complex
manifold of dimension $n \geq 3$, which is Kahler outside an
irreducible curve, then $M$ carries a balanced metric
TL;DR: In this paper, the authors classify Fano five-folds of index two as blow-ups of smooth manifolds along a smooth center, and classify them as smooth or irregular.
Abstract: We classify Fano fivefolds of index two which are blow-ups of smooth manifolds along a smooth center.
TL;DR: In this article, the notion of a log smooth degeneration is introduced, which is a logarithmic analogue of the central fiber of some kind of degenerations of complex manifolds over polydiscs.
Abstract: We introduce the notion of a log smooth degeneration, which is a logarithmic
analogue of the central fiber of some kind of degenerations of complex manifolds
over polydiscs. Under suitable conditions, we construct a natural cohomological
mixed Hodge complex on the reduction of a compact log smooth degeneration. In
particular, we obtain mixed Hodge structures on the log de Rham cohomologies and
$E_1$-degeneration of the log Hodge to de Rham spectral sequence for a certain
kind of compact reduced log smooth degenerations.
TL;DR: In this article, the authors reformulate Arakawa's theta lifting as a theta correspondence in the adelic setting and determine a commutation relation of Hecke operators satisfied by the lifting.
Abstract: T. Arakawa, in his unpublished note, constructed and studied a theta lifting from elliptic cusp forms to automorphic forms on the quaternion unitary group of signature $(1, q)$. The second named author proved that such a lifting provides bounded (or cuspidal) automorphic forms generating quaternionic discrete series. In this paper, restricting ourselves to the case of $q=1$, we reformulate Arakawa's theta lifting as a theta correspondence in the adelic setting and determine a commutation relation of Hecke operators satisfied by the lifting. As an application, we show that the theta lift of an elliptic Hecke eigenform is also a Hecke eigenform on the quaternion unitary group. We furthermore study the spinor $L$-function attached to the theta lift.
TL;DR: In this paper, the authors obtained a similar description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope, which includes the class of smooth complete complex toric varieties.
Abstract: The $K$-rings of non-singular complex projective varieties as well as quasi-toric manifolds were described in terms of generators and relations in earlier work of the author with V. Uma. In this paper we obtain a similar description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope, which includes the class of all smooth complete complex toric varieties.
TL;DR: In this paper, a complete classification for all Mobius isoparametric hypersurfaces in the unit six-sphere is established, combining with previous results, and a complete classifier for all Euclidean and Dupin hypersurface is established.
Abstract: An immersed umbilic-free hypersurface in the unit sphere is equipped with three Mobius invariants, namely, the Mobius metric, the Mobius second fundamental form and the Mobius form. The fundamental theorem of Mobius submanifolds geometry states that a hypersurface of dimension not less than three is uniquely determined by the Mobius metric and the Mobius second fundamental form. A Mobius isoparametric hypersurface is defined by two conditions that it has vanishing Mobius form and has constant Mobius principal curvatures. It is well-known that all Euclidean isoparametric hypersurfaces are Mobius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, combining with previous results, a complete classification for all Mobius isoparametric hypersurfaces in the unit six-sphere is established.
TL;DR: In this article, an analogue of the Kummer and Kummer-Artin-Schreier theories, twisting by a quadratic extension, is presented, where the argument is developed not only over a field but also over a ring, as generally as possible.
Abstract: We discuss an analogue of the Kummer and Kummer-Artin-Schreier theories, twisting by a quadratic extension. The argument is developed not only over a field but also over a ring, as generally as possible.
TL;DR: In this paper, Torsion-free affine connections with Ricci curvature tensors have been shown to preserve volume elements and have definite curvatures tensors, and they are shown to have a Ricci-convexity tensor.
Abstract: We show various examples of torsion-free affine connections which preserve volume elements and have definite Ricci curvature tensors.
TL;DR: In this article, the maximal cyclotomic extension of a finite algebraic number and its two abelian extensions, the maximal Abelian extension and the maximal ABELIAN extension with certain restricted ramification, are investigated.
Abstract: We shall consider the maximal cyclotomic extension of a finite algebraic number
field and its two abelian extensions, the maximal abelian extension and the
maximal abelian extension with certain restricted ramification. We shall
investigate the structure of these Galois groups with the action of the
cyclotomic Galois group.
TL;DR: In this paper, an integral analogue of the fiber polytope of Billera and Sturmfels is described, and the relation of H_0 to the main component of the inverse limit of GIT quotients of an affine toric variety under a torus under a subtorus is discussed.
Abstract: Let \X be an affine toric variety under a torus \T and let T be a subtorus. The general T-orbit closures in \X and their flat limits are parametrized by the main component H_0 of the toric Hilbert scheme. Further, the quotient torus \T/T acts on H_0 with a dense orbit. We describe the fan of this toric variety; this leads us to an integral analogue of the fiber polytope of Billera and Sturmfels. We also describe the relation of H_0 to the main component of the inverse limit of GIT quotients of \X by T.
TL;DR: In this article, the Euler-Poincare characteristic of homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold is computed.
Abstract: We compute the Euler-Poincare characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.
TL;DR: In this paper, the authors prove an isomorphism of Friedlander and Voevodsky motives for a smooth proper field which admits resolution of singularities in the sense of Voevodsky and Friedlander.
Abstract: Let $k$ be a perfect field which admits resolution of singularities in the sense of Friedlander and Voevodsky (for example, $k$ of characteristic $0$). Let $X$ be a smooth proper $k$-variety of pure dimension $n$ and $Y,Z$ two disjoint closed subsets of $X$. We prove an isomorphism \[ M(X-Z,Y)\simeq M(X-Y,Z)^*(n)[2n], \] where $M(X-Z,Y)$ and $M(X-Y,Z)$ are relative Voevodsky motives, defined in his triangulated category $\operatorname{DM}_{\rm gm}(k)$.
TL;DR: In this article, the Markoff transformations on the Fricke surface have been extended to the Euclidean 3-space with constant k = 4, where k is the number of tori.
Abstract: By the Fricke surfaces, we mean the cubic surfaces defined by the equation
$p^2+q^2+r^2-pqr-k=0$ in the Euclidean 3-space with the coordinates $(p,q,r)$
parametrized by constant $k$. When $k=0$, it is naturally isomorphic to the
moduli of once-punctured tori. It was Markoff who found the transformations,
called Markoff transformations, acting on the Fricke surface. The transformation
is typically given by $(p,q,r)\mapsto (r,q,rq-p)$ acting on $\boldsymbol{R}^3$
that keeps the surface invariant. In this paper we propose a way of
interpolating the action of Markoff transformation. As a result, we show that
one portion of the Fricke surface with $k=4$ admits a ${\rm
GL}(2,\boldsymbol{R})$-action extending the Markoff transformations.
TL;DR: In this article, a concrete smooth Fano 5-polytope P with 8 vertices was examined and it was shown that there is no polytope R with 9 vertices such that R contains P.
Abstract: We examine a concrete smooth Fano 5-polytope P with 8 vertices with the following properties: There does not exist a smooth Fano 5-polytope Q with 7 vertices such that P contains Q, and there does not exist a smooth Fano 5-polytope R with 9 vertices such that R contains P . As the polytope P is not pseudo-symmetric, it is a counter example to a conjecture proposed by Sato.
TL;DR: In this paper, the existence of maximal slices in anti-de-Sitter spaces (ADS) with small boundary data at spatial infinity was proved by implicit function theorem, and the main argument was carried out by implicit functions theorem and boundary behavior of totally geodesic slices in ADS spaces.
Abstract: We prove the existence of maximal slices in anti-de Sitter spaces (ADS spaces) with small boundary data at spatial infinity. The main argument is carried out by implicit function theorem. We also get a necessary and sufficient condition for the boundary behavior of totally geodesic slices in ADS spaces. Moreover, we show that any isometric and maximal embedding of hyperbolic spaces into ADS spaces must be totally geodesic. Combined with this, we see that most of maximal slices obtained in this paper are not isometric to hyperbolic spaces, which implies that the Bernstein Theorem in ADS space fails.
TL;DR: In this article, the authors construct invariant complex structures of a compact 3-symmetric space by means of the canonical almost complex structure of the underlying manifold and some involutions of a Lie group.
Abstract: We construct invariant complex structures of a compact 3-symmetric space by means of the canonical almost complex structure of the underlying manifold and some involutions of a Lie group. Moreover, by making use of graded Lie algebras and some invariant structures of affine symmetric spaces, we classify half dimensional, totally real and totally geodesic submanifolds of a compact 3-symmetric space with respect to each invariant complex structure.