arXiv research

Classifies weakly Einstein submanifolds in space forms satisfying specific equalities.

Recently Oprea gave an improved version of Chen's inequality for Lagrangian submanifolds of $\mathbb CP^n(4)$. For minimal submanifolds this inequality coincides with the original previously proved version. We consider here those non minimal 3-dimensional Lagrangian submanifolds in $\mathbb CP^3 (4)$ attaining at all p…

The paper derives Chen-Ricci inequalities for Riemannian submersions and maps.

The paper derives inequalities for Riemannian submersions and applies them to specific space forms.

Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface $Σ$, we observe that …

Establish generalized Chen inequalities for Riemannian submersions and Riemannian maps with applications.

We solve the remaining cases of the Riemann mapping problem of Escobar. Indeed, performing a suitable scheme of the barycenter technique of Bahri-Coron via the Chen's bubbles, we solve the cases left open after the work of Chen. Thus, combining our work with the ones of Almaraz, Chen, Escobar and Marques we have that e…

Paper establishes new inequality for Riemannian maps and applies it to various space forms.

In this paper, we solve the remaining cases of the boundary Yamabe problem introduced by Escobar in 1992. Indeed, using the bubbles of Brendle-Chen, which are an adaptation to manifolds with boundary of the original ones introduced by Brendle for the study of the Yamabe flow on closed Riemannian manifolds of dimension …

Study on real hypersurfaces in complex projective plane with constant mean curvature.

Study real hypersurfaces in complex space forms for an inequality involving a contact invariant.

The paper studies geometric properties of a specific type of submanifolds in Kaehler manifolds.

The paper derives optimal inequalities for bi-slant submanifolds in metallic Riemannian space forms.

We classify Lagrangian submanifolds of complex space forms, whose second fundamental form can be written in a certain way, depending on a real parameter. For some special values of this parameter, the resulting submanifolds are ideal in the sense that they realize equality in an inequality for a Chen's delta-curvature.

Study on submanifolds with specific types of factors in Kaehler manifolds.

We give a new formula for the energy functionals E_k defined by Chen-Tian, and discuss the relations between these functionals. We also apply our formula to give a new proof of the fact that the holomorphic invariants corresponding to the E_k functionals are equal to the Futaki invariant.

We study Lagrangian immersions in the nearly Kähler $\mathbb{S}^6$ which are warped product manifolds of a $1$-dimensional base and a surface. Apart from the totally geodesic ones, they are either of constant sectional curvature $\frac{1}{16}$ or they satisfy equality in Chen's inequality, in which case the immersion i…

The Chen-Yang volume conjecture is verified for knots in handlebodies with specific boundary components.

We establish some inequalities of Chen's type between certain intrinsic invariants (involving sectional, Ricci and scalar curvatures) and the squared mean curvature of submanifolds tangent to the structure vector fields of a generalized S-space-form and we discuss the equality cases of them. We apply the obtained resul…

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C^4, as well as Lagrangian submanifolds of the near…

Study Alexander invariants and cohomology jump loci in group extensions with trivial monodromy.

n this paper, we obtain a geometric inequality between the length of the second fundamental form and the length of Lee form in terms of the warping function for a CR-warped product submanifold in a locally conformal Kaehler space form. The equality case is also investigated. Furthermore, the inequality is discussed for…

Paper proves edge-connectivity equals minimum degree for graphs with non-negative curvature.

It was proved in [8,9] that every Lagrangian submanifold $M$ of a complex space form $\tilde M^{5}(4c)$ of constant holomorphic sectional curvature $4c$ satisfies the following optimal inequality: {align}\tag{A}δ(2,2)\leq \text{\small${25}{4}$} H^{2}+8c,{align} where $H^{2}$ is the squared mean curvature and $δ(2,2)$ i…

Paper confirms Chen's biharmonic conjecture for hypersurfaces in 5D.

We prove the existence of invariant almost complex structure on any positively omnioriented quasitoric orbifold. We construct blowdowns. We define Chen-Ruan cohomology ring for any omnioriented quasitoric orbifold. We prove that the Euler characteristic of this cohomology is preserved by a crepant blowdown. We prove th…

The CR $δ$-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR $δ$-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic su…

We present Chen-Ricci inequality and improved Chen-Ricci inequality for curvature like tensors. Applying our improved Chen-Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms and C-totally real submanifolds of Sasakian space forms.

The paper derives Chen inequalities for statistical submanifolds in cosymplectic manifolds.

Proves generalized Chen's conjecture for biharmonic maps on foliations.

A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …

Proves Chen-Lin conjecture for sphere scalar curvature problem.

Proves Chen's conjecture on biharmonic submanifolds in Euclidean space and space forms.

Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.

Let $M$ be an $n$-dimensional Lagrangian submanifold of a complex space form. We prove a pointwise inequality $δ(n_1,\ldots,n_k) \leq a(n,k,n_1,\ldots,n_k) \|H\|^2 + b(n,k,n_1,\ldots,n_k)c,$ with on the left hand side any delta-invariant of the Riemannian manifold $M$ and on the right hand side a linear combination o…

Analyzes string topology operations using Chen's integrals and homotopy transfer.

We present a deRham model for Chen-Ruan cohomology ring of abelian orbifolds. We introduce the notion of \emph{twist factors} so that formally the stringy cohomology ring can be defined without going through pseudo-holomorphic orbifold curves. Thus our model can be viewed as the classical description of Chen-Ruan cohom…

We classify Hopf hypersurfaces of non-flat complex space forms CP^m(4) and CH^m(-4), denoted jointly by CQ^m(4c), that are of 2-type in the sense of B. Y. Chen, via the embedding into a suitable (pseudo) Euclidean space of Hermitian matrices by projection operators. This complements and extends earlier classifications …

New geometric proof and generalization of Chen signature theorem.

Proves effective Chen ranks conjecture for Koszul modules.

We provide a direct proof for the positivity of Chen-Nester-Tung quasi-local energy with analytic reference in spherical symmetry. A hoop-type theorem for this energy is also established. Finally, the relation between Chen-Nester-Tung and Brown-York quasi-local energies will be discussed.

Period maps surjective for certain gravitational instantons.

Paper derives inequalities for submanifolds in a specific geometric space.

The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10-13, 16, 18-21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathb…

Cylindrical contact homology linked to Ehrhart polynomials and Chen-Ruan cohomology.

For biharmonic maps, there is a famous conjecture named Chen's conjecture. In later paper, Wang and Ou gave an affirmative partial answer to submersion version of Chen's conjecture. In this paper, we give an affirmative partial answer to submersion version of generalized Chen's conjecture, that is, triharmonic Riemanni…

Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.

It is known that there exist no warped product semi-slant submanifolds in Kaehler manifolds \cite{Sahin}. Recently, Chen and Garay studied pointwise-slant submanifolds of almost Hermitian manifolds in \cite{CG} and obtained many new results for such submanifolds. In this paper, we first introduce pointwise semi-slant s…