arXiv research

Study on $G_2$-structures using Laplacian coflow and solitons.

We study the Laplacian coflow and the modified Laplacian coflow of $G_2$-structures on the $7$-dimensional Heisenberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval $(-\infty,T)$, with $0<T<+\infty$. However, for the modified Laplacian coflow, we prov…

Study explores Laplacian coflow versions on Calabi-Yau 7-manifolds.

We find explicit solutions of the Laplacian coflow of $G_2-$structures on seven-dimensional almost-abelian Lie groups. Moreover, we construct new examples of solitons for the Laplacian coflow which are not eigenforms of the Laplacian and we exhibit a solution, which is not a soliton, having a bounded interval of existe…

Real analyticity proved for modified Laplacian coflow solutions.

We survey recent progress in the study of $G_{2}$-structure Laplacian coflows, that is, heat flows of co-closed $G_{2}$-structures. We introduce the properties of the original Laplacian coflow of $G_{2}$-structures as well as the modified coflow, reviewing short-time existence and uniqueness results for the modified co…

Study $G_2$-flows reducing to complex geometry flows, focusing on $G_2$-anomaly and $G_2$-Laplacian coflow.

Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.

The paper connects complex Monge-Ampère equations to $G_2$-structures on Calabi-Yau manifolds.

We study the Laplacian flow of a $\mathrm{G}_2$-structure where this latter structure is claimed to be Locally Conformal Parallel. The first examples of long time solutions of this flow with the Locally Conformal Parallel condition are given. All of the solutions are ancient and Laplacian soliton of shrinking type. The…

We modify the Laplacian coflow of co-closed G2-structures - $\frac{d}{dt}ψ=Δψ$ where $ψ$ is the closed dual 4-form of a $G_{2}$-structure $\varphi$. The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified f…

A flow from hypersymplectic to hyperkähler structures is described.

Found a new compact G2-structure on a 7-manifold.

We consider the Laplacian coflow of a $\mathrm{G}_2$-structure on warped products of the form $M^7= M^6 \times_f S^1$ with $M^6$ a compact 6-manifold endowed with an $\mathrm{SU}(3)$-structure. We give an explicit reinterpretation of this flow as a set of evolution equations of the differential forms defining the $\mat…

We consider $G_{2}$-structures on $7$-manifolds that are warped products of an interval and a six-manifold, which is either a Calabi-Yau manifold, or a nearly Kähler manifold. We show that in these cases the $G_{2}$-structures are determined by their torsion components up to a phase factor. We then study the modified L…

Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.

We prove a general result about the stability of geometric flows of "closed" sections of vector bundles on compact manifolds. Our theorem allows to prove a stability result for the modified Laplacian coflow in G2-geometry introduced by Grigorian and for the balanced flow introduced by the authors in a previous paper.

Study on G2 structures with torsion and existence of solutions.

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} ψ= - Δ_d ψ$ where $ψ$ is the dual 4-form of a $G_2$-structure $φ$ and $Δ_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_2$-structure being coclosed ($dψ=0$). We study this flow for two explicit examples of coclosed $…

The paper proves unboundedness of Donaldson-Hitchin functionals on G2 and tG2 forms.

Defines vector Laplacian on statistical manifolds.

BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.

Paper introduces magnetic Hodge Laplacian for differential forms.

Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.

The paper extends Laplacian spectra approximations to vector bundles.

Survey of Laplacian-based methods for data dimensionality reduction and embedding.

The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph $p$-Laplacian. Unlike the …

The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.

Extended bounds on small eigenvalues for pseudo-Laplacians on hyperbolic surfaces.

Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.

Study Hodge Laplacians for manifold data, improving error bounds.

Root Laplacian Eigenmaps help in spectral embedding of graphs.

New perspective on G2-structures flow from DeTurck Laplacian.

Explains BV Laplacian on half-densities in simple terms.

Study solves sub-Laplacian equivalence on a specific Heisenberg group.

A kinematic method selects the deformation Laplacian for fluid dynamics on Riemannian manifolds.

Study shows rates for Laplacian-eigenmap methods in nonparametric regression.

Study eigenvalues of p-Laplacian on quaternionic Kähler manifolds.

The paper finds new inequalities for Laplacian and biharmonic eigenvalues on manifolds.

Researchers find second-order estimates for $p$-Laplacian in RCD spaces.

Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.

For a bounded domain $Ω$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Lap…

Fractional Laplacian inverse problem solved for connection Laplacians.

Universal inequalities found for Laplacian eigenvalues on convex domains.

Universal inequalities for Laplacian eigenvalues on convex domains.

New upper bound for Neumann Laplacian eigenvalues on convex domains.

New inequalities for planar convex domains' Laplacian eigenvalues.

We investigate the existence of closed $G_2$-structures which are solitons for the Laplacian flow on nilpotent Lie groups. We obtain that seven of the twelve Lie algebras admitting a closed $G_2$-structure do admit a Laplacian soliton. Moreover, one of them admits a continuous family of Laplacian solitons which are pai…