arXiv research

New metrics produce discrete zero sets for nondegenerate harmonic forms.

The study resolves a conjecture about harmonic forms on compact manifolds.

The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.

The study finds nondegenerate harmonic 1-forms using symmetry conditions.

The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.

Study existence of harmonic 1-forms on Calabi-Yau manifolds.

New examples of Z/2 harmonic 1-forms and their branching sets are explored.

New type of harmonic Finsler manifolds created.

The study calculates harmonic functions and 1-forms on specific 4D spaces.

No nontrivial harmonic 1-forms on certain gradient Ricci solitons.

Study metric perturbations to make degenerate harmonic forms non-degenerate.

Study Dolbeault harmonic forms on Lie group quotients with specific structures.

The paper connects harmonic forms to tree maps and character varieties.

The paper studies harmonic 1-forms on specific metric measure spaces.

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature, if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We us…

The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.

Constructs harmonic 1-forms on K3-fibred Calabi-Yau 3-folds.

Constructs harmonic spinors and 1-forms on 3-manifold connected sums and torus sums.

Compactness theorem for Fueter sections yields non-zero harmonic 1-forms.

$\mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we compactify the moduli space of $\mathcal{H}-$holomorphic curves with a priori bounds on the harmonic $1-$forms.

The study characterizes harmonic spaces and their radial eigen-functions and vector fields.

Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.

Estimates for harmonic forms on a 3-Torus, proving their existence.

We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a $\mathbb{Z}/2$ harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. The local models are $\mathbb{Z}/2$ harmonic 1-forms or spinors on $\mathbb{R}^3$ that are homogeneous with respect to res…

The n-dimensional torus is uniquely characterized by specific harmonic forms.

We consider finite energy and $L^2$ differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal compleme…

Let $x:M^m\to \bar M$, with $m\geq 3$, be an isometric immersion of a complete noncompact manifold $M$ in a complete simply-connected manifold $\bar M$ with sectional curvature satisfying $-c^2\leq K_{\bar M}\leq 0$, for some constant $c$. Assume that the immersion has finite total curvature. If $c\neq 0$, assume furth…

In this paper, we prove the nonexistence of $L^2$ harmonic 1-forms on a complete super stable minimal submanifold $M$ in hyperbolic space under the assumption that the first eigenvalue $λ_1 (M)$ for the Laplace operator on $M$ is bounded below by $(2n-1)(n-1)$. Moreover, we provide sufficient conditions for minimal sub…

New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.

We prove that on a compact $n$-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue $λ$ of the Dirac operator satisfies the inequality $λ^2 \geq \frac{n-1}{4(n-2)}\inf_M Scal$. In the limiting case the universal cover of the manifold is isometric to $R\times N$ where $N…

Study shows that dimension of Dolbeault harmonic forms is not always equal to B- on certain 4-manifolds.

We give results on the following questions about a topologically tame hyperbolic 3-manifold M : 1. Does M have nonzero square-integrable harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on (1-forms on M)/Ker(d)?

This paper studies special Lagrangian submanifolds and their deformations.

Submission was withdrawn by authors. arXiv:math/0305140

Study topological properties of foliations induced by closed 1-forms on orbifolds.

The Navier-Stokes equations on certain manifolds can perform universal computation.

The paper extends Hodge-de Rham theory to higher-dimensional Sierpinski gaskets.

Study on harmonic forms on almost Hermitian 4-manifolds, calculating dimensions and invariants.

Novel singularity models for 4D harmonic forms and spinors from polytopes.

In this article, we first consider the $L^{2}$ \textit{Morse-Novikov cohomology} on a complete Riemannian manifold $M$ equipped with a parallel $1$-form which includes Vaisman manifold. Based on a vanishing theorem of $L^{2}$ \textit{Morse-Novikov cohomology}, we prove that the $L^{2}$-harmonic forms on $M$ are identic…

On a complete Calabi-Yau manifold $M$ with maximal volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon-Hein. We prove this result by proving a Liouville type theorem for harmonic $1$-forms, which follows from a new local …

We derive a correspondence between (Lorentzian) harmonic maps into the pseudosphere $S_1^2$, with appropriate regularity conditions, and certain connection 1-forms. To these harmonic maps, we associate a representation of type Weierstrass, and we apply it to construct timelike surfaces with constant mean curvature.

Let $M$ be a closed oriented manifold of dimension $n$ and $ω$ a closed 1-form on it. We discuss the question whether there exists a Riemannian metric for which $ω$ is co-closed. For closed 1-forms with nondegenerate zeros the question was answered completely by Calabi in 1969. The goal of this paper is to give an answ…

The study classifies gradient Ricci solitons with specific vector fields.

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface $S$ where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian torus of $S$ develops into a torus that splits. If the geodesic is nonseparating then the Jacobian torus of $…

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and $M\ne SO_0(2,2)/SO(2)\tm SO(2).$ Let E be any vector bundle over M, Then any E-valued $L^2$ harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps fro…

Geometric analysis proves weak KAM solutions constant under specific conditions.

Study bounds on harmonic forms in hyperbolic 3-manifolds using Thurston norm and minimal surfaces.