arXiv research

The paper explores neural scaling laws for deep operator networks, offering a theoretical foundation.

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

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.

Neural ODEs simplified using Chen-Fliess series for Rademacher complexity analysis.

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 generalized Chen's conjecture for biharmonic maps on foliations.

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

Proves Chen-Lin conjecture for sphere scalar curvature problem.

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

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

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…

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.

The paper derives inequalities for submanifolds in quaternionic Kaehler manifolds.

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

In [6] we proved Chen's inequality regarded as a problem of constrained maximum. In this paper we introduce a Riemannian invariant obtained from Chen's invariant, replacing the sectional curvature by the Ricci curvature of k-order. This invariant can be estimated, in the case of submanifolds M in space forms $\widetild…

We compute the Chen-Ruan orbifold cohomology ring of the Batyrev mirror orbifold of a smooth quintic hypersurface in 4-dimensional projective space. We identify the obstruction bundle for this example by using the Riemann bilinear relations for periods. We outline a general method of computing the Chen-Ruan ring for Ca…

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…

Classifies instantons on a specific gravitational instanton and computes partition functions.

Study on Kähler metrics on ruled surfaces, proving existence and non-existence.

We study the class of spacelike surfaces in the four-dimensional Minkowski space whose mean curvature vector at any point is a non-zero spacelike vector or timelike vector. These surfaces are determined up to a motion by eight invariant functions satisfying some natural conditions. The subclass of Chen surfaces is char…

Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.

Paper refines Chen-Cheng's estimates for Kähler metrics.

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…

New approach to proving Chen-Donaldson-Sun theorem with examples.

Researchers extend Chen, Erchenko, and Gogolev's result to more cases.

We show that Chen-Ruan cohomology is a homotopy invariant in certain cases. We introduce the notion of a T-representation homotopy, which is a stringent form of homotopy under which Chen-Ruan cohomology is invariant. We show that while hyperkahler quotients of the cotangent bundle to a complex vector space by a circle …

New gauge preserves Einstein metrics' interactions, proving rigidity on negatively curved manifolds.

Study limits of quasi-local angular momentum at infinity of gravitating systems.

The Chen-Yang volume conjecture states that the growth rate of the Turaev-Viro invariants of a compact oriented $3$-manifold determines its simplicial volume. In this paper we prove that the Chen-Yang conjecture is stable under $(2n+1,2)$-cabling.

Researchers construct explicit bundles for ALF metrics, revealing rational patching matrices for gravitational instantons.

The paper derives inequalities for Riemannian submersions and applies them to specific 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 …

New homotopy refinements for tangle invariants.

The paper derives inequalities for submanifolds in quaternion Kaehler-like statistical manifolds.

We investigate some relations concerning the first and the second Beltrami operators corresponding to the fundamental forms I, II, III of a surface in the three-dimensional Euclidean space and we study surfaces which are of finite type in the sense of B.-Y. Chen with respect to the fundamental forms II and III.

Based on the orthogonal Labastida-Mari{ñ}o-Ooguri-Vafa conjecture made by L. Chen & Q. Chen [5], we derive an infinite product formula for Chern-Simons partition functions, which generalizes the Liu-Peng's [19] recent results to the orthogonal case. Symmetry property of this new infinite product structure is also discu…

We first proved a compactness theorem of the Kähler metrics, which confirms a prediction of Chen. Then we prove several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed by Chen-Li-Wang to prove the small energy theorems of t…

In this note, we give a brief survey on some recent developments of biharmonic submanifolds. After reviewing some recent progress on Chen's biharmonic conjecture, the Generalized Chen's conjecture on biharmonic submanifolds of non-positively curved manifolds, and some classifications of biharmonic submanifolds of spher…