Sectional Curvature Research Articles

A note on holomorphic sectional curvature of a hermitian manifold

AbstractAs is well known, the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section on a Kähler manifold (Zheng, Complex differential geometry (2000)). In this article, we prove that if the holomorphic sectional curvature is half of the sectional curvature in a holomorphic plane section on a Hermitian manifold then the Hermitian metric is Kähler.

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.

The behavior of harmonic functions at singular points of $$\mathsf {RCD}$$ spaces

In this note we investigate the behavior of harmonic functions at singular points of $\mathsf{RCD}(K,N)$ spaces. In particular we show that their gradient vanishes at all points where the tangent cone is isometric to a cone over a metric measure space with non-maximal diameter. The same analysis is performed for functions with a Laplacian in $L^{N+\varepsilon}$. As a consequence we show that on smooth manifolds there is no a priori estimate on the modulus of continuity of the gradient of harmonic functions which depends only on lower bounds of the sectional curvature. In the same way we show that there is no a priori Calder\'on-Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature.

The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

Several Riemannian metrics and families of Riemannian metrics were defined on the manifold of Symmetric Positive Definite (SPD) matrices. Firstly, we formalize a common general process to define families of metrics: the principle of deformed metrics. We relate the recently introduced family of alpha-Procrustes metrics to the general class of mean kernel metrics by providing a sufficient condition under which elements of the former belong to the latter. Secondly, we focus on the principle of balanced bilinear forms that we recently introduced. We give a new sufficient condition under which the balanced bilinear form is a metric. It allows us to introduce the Mixed-Euclidean (ME) metrics which generalize the Mixed-Power-Euclidean (MPE) metrics. We unveal their link with the (u,v)-divergences and the (α,β)-divergences of information geometry and we provide an explicit formula of the Riemann curvature tensor. We show that the sectional curvature of all ME metrics can take negative values and we show experimentally that the sectional curvature of all MPE metrics but the log-Euclidean, power-Euclidean and power-affine metrics can take positive values.

Álgebras de Lie cuyos grupos de Lie tienen curvatura seccional negativa

The aim of this work is to completely describe two families of Lie algebras whose Lie groups have negative sectional curvature. The first family consists of Lie algebras satisfying the following property: given any two vectors in the Lie algebra, the linear subspace spanned by them is a Lie subalgebra. On the other hand, the second family consists of reduced Lie algebras of Iwasawa type.

New Characterizations of the Whitney Spheres and the Contact Whitney Spheres

In this paper, based on the classical Yano’s formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction \(J\mathbf {H}\) and the norm of the covariant differentiation of the second fundamental form h, where J is the almost complex structure and \(\mathbf {H}\) is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure \((\varphi ,\xi ,\eta ,g)\), we also establish an optimal integral inequality involving the Ricci curvature in the direction \(\varphi \mathbf {H}\) and the norm of the modified covariant differentiation of the second fundamental form. The integral inequality is optimal in the sense that all submanifolds attaining the equality are completely classified. As direct consequences, we obtain new and global characterizations for the Whitney spheres in complex space forms as well as the contact Whitney spheres in Sasakian space forms. Finally, we show that, just as the Whitney spheres in complex space forms, the contact Whitney spheres in Sasakian space forms are locally conformally flat manifolds with sectional curvatures non-constant.

The Wu–Yau theorem on Sasakian manifolds

In this paper, we proved that a compact Sasakian manifold [Formula: see text] with negative transverse holomorphic sectional curvature must have a Sasakian structure [Formula: see text] with negative transverse Ricci curvature. Similarly, a compact Sasakian manifold with nonpositive transverse holomorphic sectional curvature, then the negative first basic Chern class [Formula: see text] is transverse nef and we have the Miyaoka–Yau-type inequality. When transverse holomorphic sectional curvature is quasi-negative, we obtain a Chern number inequality.

CARLSON–GRIFFITHS THEORY FOR COMPLETE KÄHLER MANIFOLDS

AbstractWe investigate Carlson–Griffiths’ equidistribution theory of meormorphic mappings from a complete Kähler manifold into a complex projective algebraic manifold. By using a technique of Brownian motions developed by Atsuji, we obtain a second main theorem in Nevanlinna theory provided that the source manifold is of nonpositive sectional curvature. In particular, a defect relation follows if some growth condition is imposed.

The first eigenvalue for the p-Laplacian on Lagrangian submanifolds in complex space forms

The goal of this paper is to prove new upper bounds for the first positive eigenvalue of the [Formula: see text]-Laplacian operator in terms of the mean curvature and constant sectional curvature on Riemannian manifolds. In particular, we provide various estimates of the first eigenvalue of the [Formula: see text]-Laplacian operator on closed orientate [Formula: see text]-dimensional Lagrangian submanifolds in a complex space form [Formula: see text] with constant holomorphic sectional curvature [Formula: see text]. As applications of our main theorem, we generalize the Reilly-inequality for the Laplacian [R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52(4) (1977) 525–533] to the [Formula: see text]-Laplacian for a Lagrangian submanifold in a complex Euclidean space and complex projective space for [Formula: see text] and [Formula: see text], respectively.

Quasiprojective manifolds with negative holomorphic sectional curvature

Let (M,ω) be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu–Yau and Tosatti–Yang that M is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of M is of general type, thus confirming in this particular case a celebrated conjecture of Lang. Moreover, we can extend the theorem to the quasinegative curvature case building on earlier results of Diverio–Trapani. Finally, we investigate the more general setting of a quasiprojective manifold X∘ endowed with a Kähler metric with negative holomorphic sectional curvature, and we prove that such a manifold X∘ is necessarily of log-general type.

Visco-Da Rios Equation in 3-Dimensional Riemannian Manifold

In this paper, we study two classes of a space curve evolution in terms of Frenet frame for the visco-Da Rios equation in a 3-dimensional Riemannain manifold. Also, we obtain the connection between the visco-Da Rios equation and nonlinear Schrödinger equation for two classes in a 3-dimensional Riemannain manifold with constant sectional curvature. Finally, we give the Bäcklund transformations of space curve with the visco-Da Rios equation.

Pythagorean Isoparametric Hypersurfaces in Riemannian and Lorentzian Space Forms

We introduce the notion of a Pythagorean hypersurface immersed into an n+1-dimensional pseudo-Riemannian space form of constant sectional curvature c∈−1,0,1. By using this definition, we prove in Riemannian setting that if an isoparametric hypersurface is Pythagorean, then it is totally umbilical with sectional curvature φ+c, where φ is the Golden Ratio. We also extend this result to Lorentzian ambient space, observing the existence of a non totally umbilical model.

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Using Rauch’s comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li–Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon-type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch’s comparison theorem, the proofs mainly rely on the first variation formula and thus are valid for general varifolds.

On complete submanifolds with parallel normalized mean curvature in product spaces

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces $M^{n}(c)\times \mathbb {R}$ , where $M^{n}(c)$ is a space form with constant sectional curvature $c\in \{-1,1\}$ , it is shown. As an application is obtained rigidity results for submanifolds with constant second mean curvature.

On first and second bm$k$-Ricci curvatures for Hermitian manifold

This paper studies the basic properties of the non-positive or non-negative first and second $k$-Ricci curvatures.Firstly, we obtain the inequality for the Ricci curvature and the holomorphic sectional curvature. Secondly, we give vanishing theorems on compact Hermitian manifolds with positive first or second $k$-Ricci curvature.In particular, we prove that if a compact Kähler manifold has a Hermitian metric with a positive second $k$-Ricci curvature ($k\geq2$), then it must be a projective manifold.

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

Let C be a strictly convex domain in a three-dimensional Riemannian manifold with sectional curvature bounded above by a constant, and let $$\Sigma $$ be a constant mean curvature surface with free boundary in C. We provide a pinching condition on the length of the traceless second fundamental form on $$\Sigma $$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if C is a geodesic ball of three-dimensional space forms, then $$\Sigma $$ is either a spherical cap or a Delaunay surface.

On a submersion of generic submanifold of a nearly Kaehler manifold

The purpose of this paper is to study the submersions of a generic submanifold [Formula: see text] of a nearly Kaehler manifold [Formula: see text]. It is proved that the base space [Formula: see text] of such a submersion which is an almost Hermitian manifold, turns out to be nearly Kaehler. It is also shown under certain conditions that base [Formula: see text] becomes Kaehler. Further, we give some curvature relations; specifically, we derive the relation between the holomorphic sectional curvature of [Formula: see text] restricted to the horizontal distribution and that of [Formula: see text].

Schwarz Lemma: The Case of Equality and an Extension

We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In particular, the holomorphic map is totally geodesic and has constant rank. In the second part, we study the holomorphic sectional curvature on an almost Hermitian manifold and establish a Schwarz lemma in terms of holomorphic sectional curvatures in almost Hermitian setting.

Certain results on trans-paraSasakian 3-manifolds

Abstract Let M be a trans-paraSasakian 3-manifold. In this paper, the necessary and sufficient condition for the Reeb vector field of a trans-paraSasakian 3-manifold to be harmonic is obtained. Also, it is proved that the Ricci operator of M is invariant along the Reeb flow if and only if M is a paracosymplectic manifold, an α-paraSasakian manifold or a space of negative constant sectional curvature.

Liouville Theorems for Holomorphic Maps on Pseudo-Hermitian Manifolds

We prove some Liouville type results for generalized holomorphic maps in three classes: maps from pseudo-Hermitian manifolds to almost Hermitian manifolds, maps from almost Hermitian manifolds to pseudo-Hermitian manifolds and maps from pseudo-Hermitian manifolds to pseudo-Hermitian manifolds, assuming that the domains are compact. For instance, we show that any \((J,J^N)\) holomorphic map from a compact pseudo-Hermitian manifold M with nonnegative (resp. positive) pseudo-Hermitian sectional curvature to an almost Hermitian manifold N with negative (resp. nonpositive) holomorphic sectional curvature is constant. We also construct explicit almost CR structures on a complex vector bundle over an almost CR manifold.