Real Submanifolds Research Articles

Rigidity results on totally real submanifolds in complex space forms

Rigidity results on totally real submanifolds in complex space forms

Bergman kernel functions associated to measures supported on totally real submanifolds

Abstract We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold 𝐾 in C n \mathbb{C}^{n} is of polynomial growth. For example, this holds in dimension one if 𝐾 is a finite union of transverse Jordan arcs in ℂ. Our bounds are sharp when 𝐾 is smooth. We give an application to the equidistribution of the zeros of random polynomials, which extends a result of Shiffman–Zelditch to the higher-dimensional setting.

Common Singularities of Commuting Vector Fields

We study the singularities of commuting vector fields of a real submanifold of a Kähler manifold Z.

Chen-like Inequalities on Submanifolds of Cosymplectic 3-Space Forms

In this paper, some equalities and inequalities involving the Riemannian curvature invariants are obtained on 3-semi slant submanifolds of cosymplectic 3-space forms. Obtained relations for 3-semi slant submanifolds are examined on 3-slant, invariant and totally real submanifolds.

Properties of gradient maps associated with action of real reductive group

Let [Formula: see text] be a Kähler manifold and let [Formula: see text] be a compact connected Lie group with Lie algebra [Formula: see text] acting on [Formula: see text] and preserving [Formula: see text]. We assume that the [Formula: see text]-action extends holomorphically to an action of the complexified group [Formula: see text] and the [Formula: see text]-action on [Formula: see text] is Hamiltonian. Then there exists a [Formula: see text]-equivariant momentum map [Formula: see text]. If [Formula: see text] is a closed subgroup such that the Cartan decomposition [Formula: see text] induces a Cartan decomposition [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the Lie algebra of [Formula: see text], there is a corresponding gradient map [Formula: see text]. If [Formula: see text] is a [Formula: see text]-invariant compact and connected real submanifold of [Formula: see text] we may consider [Formula: see text] as a mapping [Formula: see text] Given an [Formula: see text]-invariant scalar product on [Formula: see text], we obtain a Morse like function [Formula: see text] on [Formula: see text]. We point out that, without the assumption that [Formula: see text] is a real analytic manifold, the Lojasiewicz gradient inequality holds for [Formula: see text]. Therefore, the limit of the negative gradient flow of [Formula: see text] exists and it is unique. Moreover, we prove that any [Formula: see text]-orbit collapses to a single [Formula: see text]-orbit and two critical points of [Formula: see text] which are in the same [Formula: see text]-orbit belong to the same [Formula: see text]-orbit. We also investigate convexity properties of the gradient map [Formula: see text] in the Abelian case. In particular, we study two-orbit variety [Formula: see text] and we investigate topological and cohomological properties of [Formula: see text].

The second fundamental form of the real Kaehler submanifolds

Abstract Let $f\colon M^{2n}\to \mathbb {R}^{2n+p}$ , $2\leq p\leq n-1$ , be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng (2013, Michigan Mathematical Journal 62, 421–441) conjectured that if the codimension is $p\leq 11$ , then, along any connected component of an open dense subset of $M^{2n}$ , the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces in the kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\mathbb {R}^{2n+p}$ of larger dimension than $2n$ . This bold conjecture was proved by Dajczer and Gromoll just for codimension 3 and then by Yan and Zheng for codimension 4. In this paper, we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step for a possible classification of the nonholomorphic Kaehler submanifolds lying with low codimension in Euclidean space. A counterexample shows that our proof does not work for higher codimension, indicating that proposing $p=11$ in the conjecture as the largest codimension is appropriate.

Explicit construction of stationary discs and its consequences for nondegenerate quadrics

We give an explicit construction of a key family of stationary discs attached to a nondegenerate model quadric in $\mathbb{C}^N$ and derive a necessary condition for which (each lift) of those stationary discs is uniquely determined by its $1$-jet at a given point via a local diffeomorphism. This unique $1$-jet determination is a crucial step to deduce $2$-jet determination for CR automorphisms of generic real submanifolds in $\mathbb{C}^N$.

On $C$-totally real submanifolds of $\mathbb{S}^{2n+1}(1)$ with non-negative sectional curvature

On $C$-totally real submanifolds of $\mathbb{S}^{2n+1}(1)$ with non-negative sectional curvature

Holomorphicity of real Kaehler submanifolds

Holomorphicity of real Kaehler submanifolds

Geometry of CR-Slant Warped Products in Nearly Kaehler Manifolds

Recently, we studied CR-slant warped products B1×fM⊥, where B1=MT×Mθ is the Riemannian product of holomorphic and proper slant submanifolds and M⊥ is a totally real submanifold in a nearly Kaehler manifold. In the continuation, in this paper, we study B2×fMθ, where B2=MT×M⊥ is a CR-product of a nearly Kaehler manifold and establish Chen’s inequality for the squared norm of the second fundamental form. Some special cases of Chen’s inequality are given.

Minimal Legendrian submanifolds in Sasakian space forms with C-parallel second fundamental form

It is known that Cheng-He-Hu [3] gave a complete classification of the n-dimensional C-totally real (or equivalently, Legendrian) minimal submanifolds in the (2n+1)-dimensional Sasakian space form N2n+1(c) with constant sectional curvature. According to this classification and the proof therein, such minimal Legendrian submanifolds are all of C-parallel second fundamental form. The purpose of this paper is to extend the above result partly, and as the main results, we classify all the minimal Legendrian submanifolds in N2n+1(c) with C-parallel second fundamental form in cases n=2,3,4.

Real Kaehler submanifolds in codimension up to four

Let f\colon M^{2n}\to\mathbb{R}^{2n+4} be an isometric immersion of a Kaehler manifold of complex dimension n\geq 5 into Euclidean space with complex rank at least 5 everywhere. Our main result is that, along each connected component of an open dense subset of M^{2n} , either f is holomorphic in \mathbb{R}^{2n+4}\cong\mathbb{C}^{n+2} , or it is in a unique way a composition f=F\circ h of isometric immersions. In the latter case, we have that h\colon M^{2n}\to N^{2n+2} is holomorphic and F\colon N^{2n+2}\to\mathbb{R}^{2n+4} belongs to the class, by now quite well understood, of non-holomorphic Kaehler submanifolds in codimension two. Moreover, the submanifold F is minimal if and only if f is minimal.

Plurisubharmonic functions and real submanifolds of [formula omitted

We give an estimate for the volume of an analytic variety (or more generally the mass of a positive closed current) close to a real submanifold M. Applications are given to the Hausdorff measure of the intersection of the variety with M and the exponential integrability of plurisubharmonic functions on M.

Sequential warped product submanifolds in nearly Kaehler manifolds

A new class of warped product manifolds which is known as sequential warped product manifolds have been defined in [15] and studied in detail in [9]. This article is dedicated to study sequential warped product submanifolds having factors holomorphic, totally real and pointwise slant submanifolds of nearly Kaehler manifolds. We obtained Chen?s inequality for sequential warped product submanifolds involving second fundamental form and warping functions.

Convergence of the Fefferman-Graham expansion and complex black hole anatomy

Given a set of sources and one-point function data for a Lorentzian holographic QFT, does the Fefferman-Graham expansion converge? If it does, what sets the radius of convergence, and how much of the interior of the spacetime can be reconstructed using this expansion? As a step towards answering these questions we consider real analytic conformal field theory data, where in the absence of logarithms, the radius is set by singularities of the complex metric reached by analytically continuing the Fefferman-Graham radial coordinate. With the conformal boundary at the origin of the complex radial plane, real Lorentzian submanifolds appear as piecewise paths built from radial rays and arcs of circles centred on the origin. This allows singularities of Fefferman-Graham metric functions to be identified with gauge-invariant singularities of maximally extended black hole spacetimes, thereby clarifying the physical cause of the limited radius of convergence in such cases. We find black holes with spacelike singularities can give a radius of convergence equal to the horizon radius, however for black holes with timelike singularities the radius is smaller. We prove that a finite radius of convergence does not necessarily follow from the existence of an event horizon, a spacetime singularity, nor from caustics of the Fefferman-Graham gauge, by providing explicit examples of spacetimes with an infinite radius of convergence which contain such features.

On the topology of real Lagrangians in toric symplectic manifolds

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.

Topological invariants and Holomorphic Mappings

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the k-sphere into the manifold are investigated. The invariants are monotonic under holomorphic mappings and strictly monotonic under certain circumstances. Applications to holomorphic maps of annular regions in ℂ and tubular neighborhoods of compact totally real submanifolds in general in ℂ n , n≥2, are given. The contractibility of a hyperbolic domain with contracting holomorphic mapping is explained.

On C-totally real minimal submanifolds of the Sasakian space forms with parallel Ricci tensor

On C-totally real minimal submanifolds of the Sasakian space forms with parallel Ricci tensor

On Beloshapka’s rigidity conjecture for real submanifolds in complex space

A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length $l \geq 3$ of their Levi–Tanaka algebra are rigid, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the complex tangent space. For the length $l = 3$, Beloshapka’s Conjecture was proved by Gammel and Kossovskiy in 2006. In this paper, we prove the Conjecture for arbitrary length $l \geq 3$. As another application of our method, we construct polynomial models of length $l \geq 3$, which are not totally nondegenerate and admit large groups of point preserving nonlinear automorphisms.

P-harmonic 1-forms on totally real submanifolds in complex space forms

In this paper, we consider an n-dimensional totally real submanifold M in complex space forms. We establish some vanishing type theorems for p-harmonic 1-forms on such submanifolds in both cases: either M is minimal; or M is non-minimal.