Real Hypersurfaces Research Articles

Ricci-Bourguignon solitons on real hypersurfaces in the complex hyperbolic quadric

Ricci-Bourguignon solitons on real hypersurfaces in the complex hyperbolic quadric

Transversal Killing operators on real hypersurfaces in nonflat complex space forms

In this paper, we prove that the structure Lie operator on a real hypersurface in a nonflat complex space form is transversally Killing if and only if so is the structure Jacobi operator. One of these two conditions holds if and only if the hypersurface is of type (A).

Extrinsic circular trajectories on real hypersurfaces of type (A2) in a complex projective space

A smooth curve parameterized by its arclength on a submanifold is said to be extrinsic circular if it is a circle as a curve on the ambient space. We study congruence classes of extrinsic circular trajectories for magnetic fields associated with the contact metric structure on a real hypersurface of type (A2) in a complex projective space classifying them by their extrinsic geodesic curvatures and by complex torsions.

The $\mathfrak{A}$-principal real hypersurfaces in complex quadrics

The $\mathfrak{A}$-principal real hypersurfaces in complex quadrics

The equivalence theory for infinite type hypersurfaces in ℂ²

We develop a classification theory for real-analytic hypersurfaces in C 2 \mathbb {C}^{2} in the case when the hypersurface is of infinite type at the reference point. This is the remaining, not yet understood case in C 2 \mathbb {C}^{2} in the Problème local, formulated by H. Poincaré in 1907 and asking for a complete biholomorphic classification of real hypersurfaces in complex space. One novel aspect of our results is a notion of smooth normal forms for real-analytic hypersurfaces. We rely fundamentally on the recently developed CR-DS technique in CR-geometry.

Systoles and Lagrangians of random complex algebraic hypersurfaces

Let n\geq 1 be an integer, and \mathcal L \subset \mathbb{R}^n be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exist c>0 and d_0\geq 1 such that for any d\geq d_0 , any smooth complex projective hypersurface Z in \mathbb{C} P^n of degree d contains at least c\dim H_{*}(Z, \mathbb{R}) disjoint Lagrangian submanifolds diffeomorphic to \mathcal L , where Z is equipped with the restriction of the Fubini–Study symplectic form (Theorem 1.1). If moreover all connected components of \mathcal L have non-vanishing Euler characteristic, which implies that n is odd, the latter Lagrangian submanifolds form an independent family in H_{n-1}(Z, \mathbb{R}) (Corollary 1.2). These deterministic results are consequences of a more precise probabilistic theorem (Theorem 1.23) inspired by a 2014 result by J.-Y. Welschinger and the author on random real algebraic geometry, together with quantitative Moser-type constructions (Theorem 3.4). For n=2 , the method provides a uniform positive lower bound for the probability that a projective complex curve in \mathbb{C} P^2 of given degree equipped with the restriction of the ambient metric has a systole of small size (Theorem 1.6), which is an analog of a similar bound for hyperbolic curves given by M. Mirzakhani (2013). In higher dimensions, we provide a similar result for the (n-1) -systole introduced by M. Berger (1972) (Corollary 1.14). Our results hold in the more general setting of vanishing loci of holomorphic sections of vector bundles of rank between 1 and n tensored by a large power of an ample line bundle over a projective complex n -manifold (Theorem 1.20).

Multiplicity, regularity and blow-spherical equivalence of real analytic sets

This article is devoted to studying multiplicity and regularity of analytic sets. We present an equivalence for analytic sets, named blow-spherical equivalence, which generalizes differential equivalence and subanalytic bi-Lipschitz equivalence and, with this approach, we obtain several applications on analytic sets. On multiplicity, we present a generalization for Gau–Lipman’s Theorem about differential invariance of the multiplicity in the complex and real cases, and we show that the multiplicity \(\mathrm{mod}\,2\) is invariant under blow-spherical homeomorphisms in the case of real analytic curves and surfaces and also for a class of real analytic foliations and is invariant by (image) arc-analytic blow-spherical homeomorphisms in the case of real analytic hypersurfaces, generalizing some results proved by G. Valette. On regularity, we show that blow-spherical regularity of real analytic sets implies \(C^1\) smoothness only in the case of real analytic curves. We present also a complete classification of the germs of real analytic curves.

On Levi flat hypersurfaces with transversely affine foliation

We prove the non-existence of real analytic Levi flat hypersurface whose complement is 1-convex and Levi foliation is transversely affine in a compact Kähler surface.

The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds

We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a K\"ahler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac12$-positivity in the sense of Cao, Chang, and Chen, for compact "convex" real hypersurfaces in a K\"ahler manifold of nonnegative holomorphic sectional curvature. Our approach is rather simple and uses a version of the Gauss equation for semi-isometric CR immersions of pseudohermitian manifolds into K\"ahler manifolds.

Some New Results on Trans‐Sasakian Manifolds

In this paper, we classify trans‐Sasakian manifolds which are realized as real hypersurfaces in a complex space form. We also investigate trans‐Sasakian manifolds whose Reeb vector fields are harmonic‐Killing. The above results bring some new characterizations for the property of trans‐Sasakian 3‐manifolds.

Structure Lie operator on real hypersurfaces of complex two-plane Grassmannians

It is proved that on a real hypersurface $M$ in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, the structure Lie operator is parallel if and only if $M$ is an open part of a tube around a totally geodesic $G_2(\mathbb {C}^{m+1})$ in $G_2(\ma

Sums of CR and projective dual CR functions

A smooth, strongly $\mathbb{C}$-convex, real hypersurface $S$ in $\mathbb{CP}^n$ admits a projective dual CR structure in addition to the standard CR structure. Given a smooth function $u$ on $S$, we provide characterizations for when $u$ can be decomposed as a sum of a CR function and a dual CR function. Following work of Lee on pluriharmonic boundary values, we provide a characterization using differential forms. We further provide a characterization using tangential vector fields in the style of Audibert and Bedford.

On the geometry of the symmetrized bidisc

We study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $\mathbb{G}$ on itself. The automorphism group is 3 real dimensional. It foliates $\mathbb{G}$ into leaves all of which are 3 real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaev's classification of all Kobayashi-hyperbolic 2 complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension 3 studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domain \[\{(z_1,z_2)\in \mathbb{C} ^2 : 1+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|, Im(z_1 (1+\overline{z_2}))>0\}\] in Isaev's list. Isaev calls it $\mathcal D_1$. The road to the biholomorphism is paved with various geometric insights about $\mathbb{G}$. Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of $\mathcal D_1$. Among the results on $\mathcal D_1$, of particular interest is the fact that $\mathcal D_1$ is a symmetrization. When we symmetrize (appropriately defined in the context in the last section) either $\Omega_1$ or $\mathcal{D}^{(2)}_1$ (Isaev's notation), we get $\mathcal D_1$. These two domains $\Omega_1$ and $\mathcal{D}^{(2)}_1$ are in Isaev's list and he mentioned that these are biholomorphic to $\mathbb{D} \times \mathbb{D}$. We produce explicit biholomorphisms between these domains and $\mathbb{D} \times \mathbb{D}$.

Semisymmetric hypersurfaces in complex hyperbolic two-plane Grassmannians

In this paper, we introduce new notions of symmetric operators such as semisymmetric shape operator and structure Jacobi operator in complex hyperbolic two-plane Grassmannians. Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$ with such notions.

Real Hypersurfaces in Complex Grassmannians of Rank Two

It is known that there does not exist any Hopf hypersurface in complex Grassmannians of rank two of complex dimension 2m with constant sectional curvature for m≥3. The purpose of this article is to extend the above result, and without the Hopf condition, we prove that there does not exist any locally conformally flat real hypersurface for m≥3.

A certain η-parallelism on real hypersurfaces in a nonflat complex space form

Abstract In this paper, we give the complete classification of real hypersurfaces in a nonflat complex space form from the viewpoint of the η-parallelism of the tensor field h(= (1/2)𝓛 ξ ϕ). In addition we investigate real hypersurfaces whose tensor h is either Killing type or transversally Killing tensor. In particular, we shall determine Hopf hypersurfaces whose tensor h is transversally Killing tensor by using an application of the classification of real hypersurfaces admitting η-parallelism with respect to the tensor h.

Yamabe and gradient Yamabe solitons on real hypersurfaces in the complex quadric

In this paper, we give a complete classification of Yamabe solitons and gradient Yamabe solitons on real hypersurfaces in the complex quadric [Formula: see text]. In the following, as an application, we show a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on Hopf real hypersurfaces in the complex quadric [Formula: see text].

Multiplicity, regularity and Lipschitz geometry of real analytic hypersurfaces

Multiplicity, regularity and Lipschitz geometry of real analytic hypersurfaces

Real hypersurfaces in the complex hyperbolic quadric with Reeb invariant Ricci tensor

Real hypersurfaces in the complex hyperbolic quadric with Reeb invariant Ricci tensor

Real Hypersurfaces in the Complex Projective Plane Satisfying an Equality Involving $\delta(2)$

It was proved in Chen's paper [3] that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$\delta(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and $\delta(2)$ is a $\delta$-invariant introduced by him. In this paper, we study non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field. We describe how to obtain all such hypersurfaces.