Minimal 2-spheres in a complex projective space

From the Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is the classiflcation of Lagrangian submanifolds with parallel second fundamental form. In 1980’s, H. Naitoh completely classifled the Lagrangian submanifolds with parallel second fundamental form and without Euclidean factor in complex projective space, by using the theory of Lie groups and symmetric spaces. He showed that such a submanifold is always locally symmetric and is one of the symmetric spaces: SO(k + 1)=SO(k)(k ‚ 2), SU(k)=SO(k)(k ‚ 3), SU(k)(k ‚ 3), SU(2k)=Sp(k)(k ‚ 3), E6=F4. In this paper, we completely classify the Lagrangian submanifolds in complex projective space with parallel second fundamental form by an elementary geometrical method. We prove that such a Lagrangian submanifold is either totally geodesic or the Calabi product of a point with a lower dimensional Lagrangian submanifold with parallel second fundamental form, or the Calabi product of two lower dimensional Lagrangian submanifolds with parallel second fundamental form, or one of the standard symmetric spaces: SU(k)=SO(k)(k ‚ 3), SU(k)(k ‚ 3), SU(2k)=Sp(k)(k ‚ 3), E6=F4. As the arguments are of a local nature, at the same time, due to the correspondence between C-parallel Lagrangian submanifolds in Sasakian space forms and parallel Lagrangian submanifolds in complex space forms, we can also give a complete classiflcation of all Cparallel submanifolds of S 2n+1 equipped with its standard Sasakian structure. 2000 Mathematics Subject Classiflcation: Primary 53B25; Secondary 53C42

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.

Let $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ be a K\"ahler manifold obtained by blowing up a complex projective space $\mathbb{P}^n$ along a line $\mathbb{P}^1$. We prove that $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ does not admit constant scalar curvature K\"ahler metrics in any rational K\"ahler class, but admits extremal metrics, with an explicit formula in action-angle coordinates, in K\"ahler classes that are close to the pullback of the Fubini--Study class.

In the case where $\Sigma$ is a Riemann sphere, the above result was obtained by Siu [S-l, $Z$] (see also [B-R-S]). In the case where $M$ is a complex projective space and deg $f|\geqq m(p-1)/(m+1)$ where $p$ denotes the genus of $\Sigma$ , the above result was obtained by Eells and Wood [E-W]. They used algebraic geometric arguments (theorems of Riemann-Roch and Grothendieck). Recently, by using a twistor space over the domain manifold, Burns and de Bartolomeis have shown the above result in the case where $M$ is a complex projective space (cf. Remark 6 of [B-R-S]). We show the above theorem by a simple argument for the second variations used in [L-S].

We study holomorphic immersions f:X →M from a complex manifoldX into a Kahler manifold of constant holomorphic sectional curvatureM, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. ForX compact we show that the tangent sequence splits holomorphically if and only iff is a totally geodesic immersion. ForX not necessarily compact we relate an intrinsic cohomological invariantp(X) onX, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariantsp(X) and?(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially whenX is a complex surface andM is of complex dimension 4, under the assumption thatX admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.

The concept of Brody hyperbolicity is interpreted in terms of homotopy theoretic structures. We extend the definition of Brody hyperbolicity to simplicial sheaves of sets over the site of complex spaces with the strong topology. Imitating one possible definition of homotopy groups for a topological space, we defined the holotopy groups for a simplicial sheaf and showed that their vanishing in “positive” degrees is a necessary condition for a sheaf to be Brody hyperbolic. A partial converse to this theorem is proved at the end of the paper. We deduce that if X is a complex space with a nonzero holotopy group in positive degree, then X cannot be weakly equivalent (in a particular sense) to a hyperbolic complex space (in particular is not itself hyperbolic). We finish the manuscript by applying these results along with a topological realization functor, constructed in the previous section, to prove that complex projective spaces cannot be weakly equivalent to hyperbolic complex spaces.

A real hypersurface M in a complex projective space inherits an almost contact metric structure from the Kählerian structure of the ambient space. This almost contact metric structure allows us to define, for any nonzero real number k, the so-called k-th generalized Tanaka–Webster connection. With this connection and the Levi-Civita one we can associate two tensors of type (1,2) to the structure Jacobi operator R_{xi } of M. We classify real hypersurfaces in complex projective space for which such tensors satisfy a cyclic property.

In Cogdell et al., LMS Lecture Notes Series 459, 393–427 (2020), the authors proved an analogue of Kronecker’s limit formula associated to any divisor D \mathcal D which is smooth in codimension one on any smooth Kähler manifold X X . In the present article, we apply the aforementioned Kronecker limit formula in the case when X X is complex projective space C P n \mathbb {C}\mathbb {P}^n for n ≥ 2 n \geq 2 and D \mathcal D is a hyperplane, meaning the divisor of a linear form P D ( z ) P_D({z}) for z = ( Z j ) ∈ C P n {z} = (\mathcal {Z}_{j}) \in \mathbb {C}\mathbb {P}^n . Our main result is an explicit evaluation of the Mahler measure of P D P_{D} as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the L 2 L^{2} -norm of the vector of coefficients of P D P_{D} .

Lie derivatives and structure Jacobi operator on real hypersurfaces in complex projective spaces

We study the high-dimensional asymptotics of the expected number of critical points of a given Morse index of Gaussian random holomorphic sections over complex projective space. We explicitly compute the exponential growth rate of the expected number of critical points of the largest index and of diverging indices at various rates as well as the exponential growth rate for the expected number of critical points (regardless of index). We also compute the distribution of the critical values for the expected number of critical points of smallest index.

On a real hypersurface M in a complex projective space, we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka–Webster connection. Therefore, we can also consider their associated Lie derivatives. We classify real hypersurfaces such that both the Lie derivatives associated with the Levi-Civita connection and the kth g-Tanaka–Webster connection either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the shape operator of M.

This paper gives a classification of real hypersurfaces in a complex projective space under assumptions that the structure vector $\xi$ is principal, the focal map has constant rank, and ${\nabla _\xi }S = 0$, where S is the Ricci tensor of the real hypersurface.

The study of homogeneous and evenly homogeneous complex supermanifolds of dimension whose reduction is the complex projective space is continued. All non-split supermanifolds of this type with retract admitting the full projective group as group of automorphisms are classified. The cohomology with values in the tangent sheaf is calculated for homogeneous supermanifolds satisfying the above-mentioned conditions.

We compute the Morse index of a critical submanifold of the energy functional on the loop space of a manifold with constant sectional curvature. The case of constant non-positive sectional curvature is a known result and the case of a sphere has been proved by Klingenberg. We adapt Klingenberg’s proof of the case of a sphere to the case of constant sectional curvature, to obtain the possible Morse indices of critical submanifolds of the energy functional.

In this paper, we show the first and second variational formulas of biharmonic maps and bi-Yang-Mills fields, and show the first variation formula of k-harmonic maps, and also give an overview of our recent results in (12), i.e., classifications of all biharmonic isopara- metric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces, answers in case of bounded geometry to Chen's conjecture or Caddeo, Montaldo and Piu's one on biharmonic maps into a space of non positive curvature and the isolation phenomena of bi-Yang-Mills fields.