Dimensional Complex Projective Space Research Articles

THE NONLINEAR MASLOV INDEX AND THE CALABI HOMOMORPHISM

In this paper, we find that the asymptotic nonlinear Maslov index defined on the universal cover of the group of all contact Hamiltonian diffeomorphisms of the standard (2n - 1)-dimensional contact sphere is a quasimorphism. Then we show our main result: Let M be standard (n - 1)-dimensional complex projective space. We prove that the value of the pullback of the asymptotic nonlinear Maslov index to the universal cover of the group of Hamiltonian diffeomorphisms of M, when evaluated on a diffeomorphism supported in a sufficiently small open subset of M, equals [Formula: see text] times the Calabi invariant of this diffeomorphism.

LAMINATIONS DANS LES ESPACES PROJECTIFS COMPLEXES

We prove holomorphic immersion theorems in a finite dimensional complex projective space for kahlerian non-compact manifolds and for laminations by complex manifolds that carry a line bundle of positive curvature. In particular, we prove that on a Riemann surfaces lamination of a compact space, the space of meromorphic functions separates points if and only if every foliation cycle is non homologous to 0.

Emergent AdS3and BTZ black hole from weakly interacting hot 2d CFT

We investigate emergent holography of weakly coupled two-dimensional hyperK\"ahler sigma model on cotangent bundle of (N-1)-dimensional complex projective space at zero and finite temperature. The sigma model is motivated by the spacetime conformal field theory dual to the near-horizon geometry of Q1 D1-brane bound to Q5 D5-brane wrapped on four-torus times circle, where N = Q1*Q5. The sigma model admits nontrivial instanton for all N greater than or equal to 2, which serves as a local probe of emergent holographic spacetime. We define emergent geometry of the spacetime as that of instanton moduli space via Hitchin's information metric. At zero temperature, we find that emergent geometry is AdS3. At finite temperature, time-periodic instanton is mappable to zero temperature instanton via conformal transformation. Utilizing the transformation, we show that emergent geometry is precisely that of the non-extremal, non-rotating BTZ black hole.

Surfaces in $S^{n}$ with Prescribed Gauss Map

Let $G$ be a $C^{\infty }$-mapping from a connected Riemann surface $M$ into the complex quadric $Q_{n-1}$ in the $n$-dimensional complex projective space. We give a condition for the existence of a surface in the $n$-dimensional Euclidean unit sphere $S^{n}$ such that the Gauss map is $G$. Under this condition, if $M$ is a torus, there exists a surface in $S^{n}$ such that the Gauss map is $G$. We also show that for a connected Riemann surface $M$ there exists an immersion $X:M\rightarrow RP^{n}$ such that a neighborhood of each point of $X(M)$ is covered by a surface in $S^{n}$ with prescribed Gauss map $G$ where $RP^{n}$ is the $n$-dimensional real projective space.

On Integral Geometry in the Four Dimensional Complex Projective Space

On Integral Geometry in the Four Dimensional Complex Projective Space

On a holomorphic curve extremal for the defect relation

Let $f$ be a transcendental holomorphic curve from the complex plane into the two dimensional complex projective space of which defect relation over a set $X$ in $N$-subgeneral position is extremal. Then, there are $N-1$ vectors in $X$ whose deficiency with respect to $f$ is equal to 1.

Holomorphic Functions on (Generalised) Loop Spaces

Let ℙ n denote n dimensional complex projective space. Given k = 0,1,..., ∞, a map χ0 : S¹ -> ℙ n of class Ck, and a closed subset A ⊂ S¹ consider the space Y of those C k maps χ : S¹ -> ℙ n that agree with χ₀ along A . The space Y can be endowed with the structure of an infinite dimensional complex manifold, and our main result identifies the space of holomorphic functions on Y.

Configuration Space Models for Spaces of Maps from a Riemann Surface to Complex Projective Space

Let \mathrm{Map}_d^∗(\mathrm M_g, ℂ\mathrm{P}^{n−1}) denote the space consisting of all basepoint preserving continuous maps of degree d from a compact Riemann surface \mathrm M_g of genus g into an (n − 1) -dimensional complex projective space ℂ\mathrm{P}^{n−1} . In this paper, we construct a finite dimensional configuration space model \mathrm{SP}_n^d(\mathrm M'_g) for the infinite dimensional space \mathrm{Map}_d^∗(\mathrm M_g, ℂ\mathrm{P}^{n−1}) and show that the Atiyah–Jones type theorem (cf. [1], [12]) holds for this model.

Grassmann structures and integrable systems

We consider some natural connections which arise between right-flat ( p, q) paraconformal structures and integrable systems. We find that such systems may be formulated in Lax form with a “Lax p-tuple” of linear differential operators, depending a spectral parameter which lives in ( q−1)-dimensional complex projective space. Generally, the differential operators contain partial derivatives with respect to the spectral parameter.

Lagrangian submanifolds of the three dimensional complex projective space

We investigate Lagrangian submanifolds of the 3-dimensional complex projective space. In case the second fundamental form takes a special form, we obtain several classification theorems. As a consequence we obtain several new examples of 3-dimensional Lagrangian submanifolds.

Lagrangian submanifolds of the three dimensional complex projective space

We investigate Lagrangian submanifolds of the 3-dimensional complex projective space. In case the second fundamental form takes a special form, we obtain several classification theorems. As a consequence we obtain several new examples of 3-dimensional Lagrangian submanifolds.

Transformations birationnelles quadratiques de l'espace projectif complexe à trois dimensions

We classify birational quadratic transformations of the three dimensional complex projective space, up to linear isomorphisms of source and target. They are of three kinds, depending on the degree of the inverse, which can be 2,3 or 4. There are 30 different types; in 1871, L. Cremona described 23 of them already.

On Modified Mabuchi Functional and Mabuchi Moduli Space of Kähler Metrics on Toric Bundles

Mabuchi introduced the Mabuchi functional in [Mb1], and it turns out that it is very useful for dealing with Kahler metrics with constant scalar curvatures on compact manifolds (see [BM], etc.). One can also expect that the existence of Kahler metric with constant scalar curvature is almost equivalent to the existence of a lower bound of the Mabuchi functional (see, e.g., [Ti]). But for the case of extremal metrics, the Mabuchi functional is not applicable. Therefore, we need a new (or a modified) functional for metrics which are invariant under a maximal compact connected subgroup K of Aut(M). We did not obtain this functional until the appearing of [FM] (while we were reviewing [FM] in 1995). Mabuchi also found this functional independently [Mb3] (see also [Sm]). A definition of this functional was given in [GC]. We shall give some results and applications in this paper. It turns out that our modified Mabuchi functional M(ω1, ω2) has the property that M(ω1, g∗ω2) = M(ω1, ω2), for any g ∈ CKC(K), where CKC(K) is the centralizer of K in the complexification K of K. Moreover, the extremal metrics are exactly the local minimal points of this functional. Therefore, we expect that the existence of an extremal metric is almost equivalent to the existence of a lower bound of this functional. Surprising enough that the first application of this functional is not the existence but the uniqueness of extremal metrics on smooth toric varieties, i.e., smooth Kahler manifolds with an open (C∗)n-orbit. Therefore, there is for example at most one extremal metric in any Kahler class of the manifold obtained by blowing up two points or three points of a two dimensional complex projective space. To have the uniqueness, we consider the Mabuchi moduli space of the Kahler metrics on the toric varieties (see [Mb2], which was rediscovered by Semmes [Se1] and Donaldson [Ch]). It turns out that the moduli space is flat in this situation (see also [Se1,2]). Moreover, for any two Kahler metrics there is a unique geodesic

Differential topology of numerical range

The numerical range of an n × n matrix, also known as its field of values, is reformulated as the image of a smooth quadratic mapping from the n − 1 dimensional complex projective space to the complex plane. This paper investigates the numerical range from the perspective of differential topology (Morse theory). More specifically, the boundary of the range is interpreted as a rank 1 critical value curve and its sharp points are interpreted as rank 0 critical values. More importantly, the map is shown to have additional critical value curves in the interior of the numerical range. These additional curves are shown to have such singularity phenomena as cusps and swallow tails, to be the caustic envelopes of families of lines, and to exhibit the so-called “normal bifurcation” when an eigenvalue becomes unitarily decoupled.

The rigidity for real hypersurfaces in a complex projective space

The purpose of this paper is to give a rigidity theorem for real hypersurfaces in Pn(C) satisfying a certain geometric condition. Introduction. Let Pn{C) denote an n{ > 2)-dimensional complex projective space with the metric of constant holomorphic sectional curvature 4c. We proved in [4] that two isometric immersions of a {In — l)-dimensional Riemannian manifold M into Pn(C) are congruent if their second fundamental forms coincide. In general, the type number is defined as the rank of the second fundamental form. In this paper we shall give another rigidity theorem of the same type: THEOREM A. Let M be a {In — \)-dimensίonal Riemannian manifold, and i and ϊ be two isometric immersions of M into Pn{C) {n > 3). Assume that i and ΐ have a principal direction in common at each point of M, and that the type number of (M, ί) or (M, ΐ) is not equal to 2 at each point of M. Then i and i are congruent, that is, there is a unique isometry φ of Pn{C) such that φoi = i. We shall say that an isometry φ of a real hypersurface M in Pn{C) is principal if for each point p of M there exists a principal vector v at p such that the vector φ*{v) is also principal at φ{p), where φ^ denotes the differential of φ at p. Then as an application of Theorem A we have: THEOREM B. Let M be a homogeneous real hypersurface in Pn{C) {n>3). Assume that each isometry of M is principal. Then M is an orbit under an analytic subgroup of the projective unitary group PU{n+ 1). Note that all orbits in Pn{C) under analytic subgroups of the projective unitary group PU{n+ 1) are completely classified in [4]. The authors would like to express their thanks to the referee for his useful advice. 1. Preliminaries. Let M be a {2n— l)-dimensional Riemannian manifold, and i be 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40; Secondary 53C15.

POINCARÉ FUNCTIONS OF POINT SETS IN PROJECTIVE SPACES A COMPUTER EXPERIMENT

Let s and m(≥s+2)be two natural numbers, and Xs,m be the space of m-point sets in the (s - l)-dimensional complex projective space. We give algorithms to find the Poincare functions of the spaceXs,m.equipped with the natural projective structure, and determine them for small s amd m. The results for (s, m) = (3, 7), (3, 8), (3, 9), (4, 8) are new.

A rigidity for real hypersurfaces in a complex projective space

The purpose of this paper is to give a rigidity theorem for real hypersurfaces in Pn(C) satisfying a certain geometric condition. Introduction. Let Pn{C) denote an n{ > 2)-dimensional complex projective space with the metric of constant holomorphic sectional curvature 4c. We proved in [4] that two isometric immersions of a {In — l)-dimensional Riemannian manifold M into Pn(C) are congruent if their second fundamental forms coincide. In general, the type number is defined as the rank of the second fundamental form. In this paper we shall give another rigidity theorem of the same type: THEOREM A. Let M be a {In — \)-dimensίonal Riemannian manifold, and i and ϊ be two isometric immersions of M into Pn{C) {n > 3). Assume that i and ΐ have a principal direction in common at each point of M, and that the type number of (M, ί) or (M, ΐ) is not equal to 2 at each point of M. Then i and i are congruent, that is, there is a unique isometry φ of Pn{C) such that φoi = i. We shall say that an isometry φ of a real hypersurface M in Pn{C) is principal if for each point p of M there exists a principal vector v at p such that the vector φ*{v) is also principal at φ{p), where φ^ denotes the differential of φ at p. Then as an application of Theorem A we have: THEOREM B. Let M be a homogeneous real hypersurface in Pn{C) {n>3). Assume that each isometry of M is principal. Then M is an orbit under an analytic subgroup of the projective unitary group PU{n+ 1). Note that all orbits in Pn{C) under analytic subgroups of the projective unitary group PU{n+ 1) are completely classified in [4]. The authors would like to express their thanks to the referee for his useful advice. 1. Preliminaries. Let M be a {2n— l)-dimensional Riemannian manifold, and i be 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40; Secondary 53C15.

Totally Real Minimal Submanifolds of a Complex Projective Space

An $n$-dimensional positively curved compact totally real minimal submanifold of an $n$-dimensional complex projective space is totally geodesic.

A Remark on the Segal-Becker Theorem

Let CP°° be the infinite dimensional complex projective space and BU the classifying space of stable complex vector bundles. Then there is the natural inclusion j: CP°°—>BU and the structure map of the infinite loop space structure defined by the Bott periodicity ^:d(BU) >BU where Q,( ) =ColimllO 2( ). Let Z: Q(CP~) > BU be the composition £°Q(j). The results of Segal [8] and Becker [2] show us that there exists a map s: BU >Q^(CP°°) such that Aos = id. The main result of this paper is to show that one can take s satisfying that SQJ~inclusion : CP°° >Q(CP°°}. To show this, we will use the results of Brumfiel-Madsen [4] for the evaluation of the transfer map.

Totally Real Minimal Submanifolds in a Complex Projective Space

We give a pinching theorem with respect to the scalar curvatures of $4$-dimensional conformally flat totally real minimal submanifolds in a $4$-dimensional complex projective space.