Hodge De Rham Laplacian Research Articles

Hodge-de Rham Laplacian and geometric criteria for gravitational waves

The curvature tensor \(\hat{R}\) of a manifold is called harmonic, if it obeys the condition \(\Delta^{\text{(HR)}}\hat{R}=0\), where \(\Delta^{\text{(HR)}}=DD^{\ast} +
 D^{\ast}D\) is the Hodge–deRham Laplacian. It is proved that all solutions of the Einstein equations in vacuum, as well as all solutions of the Einstein–Cartan theory in vacuum have a harmonic curvature. The statement that only solutions of Einstein’s equations of type \(N\) (describing gravitational radiation) are harmonic is refuted.

Results Related to the Chern–Yamabe Flow

Let $$(X,\omega _0)$$ be a compact complex manifold of complex dimension n endowed with a Hermitian metric $$\omega _0$$ . The Chern–Yamabe problem is to find a conformal metric of $$\omega _0$$ such that its Chern scalar curvature is constant. As a generalization of the Chern–Yamabe problem, we study the problem of prescribing Chern scalar curvature. We then estimate the first nonzero eigenvalue of Hodge–de Rham Laplacian of $$(X,\omega _0)$$ . On the other hand, we prove a version of conformal Schwarz lemma on $$(X,\omega _0)$$ . All these are achieved by using geometric flows related to the Chern–Yamabe flow. Finally, we prove the backwards uniqueness of the Chern–Yamabe flow.

$$L^p$$-Estimates for the Heat Semigroup on Differential Forms, and Related Problems

We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let $$\overrightarrow{\Delta }_k $$ be the Hodge–de Rham Laplacian on differential k-forms with $$k \ge 1$$. By the Bochner decomposition formula, $$\overrightarrow{\Delta }_k = \nabla ^* \nabla + R_k$$, where $$\nabla $$ denotes the Levi-Civita connection and $$R_k$$ is a symmetric section of $$\mathrm{End}(\Lambda ^kT^*M)$$. Under the assumption that the negative part $$R_k^-$$ is in an enlarged Kato class, we prove that for all $$p \in [1, \infty ]$$, $$\Vert e^{-t\overrightarrow{\Delta }_k}\Vert _{p-p} \le C ( t \log t)^{\frac{D}{4}(1- \frac{2}{p})}$$ (for large t), where D is a homogeneous “dimension” appearing in the volume doubling property. This estimate can be improved if $$R_k^-$$ is strongly sub-critical. In general, $$(e^{-t\overrightarrow{\Delta }_k})_{t>0}$$ is not uniformly bounded on $$L^p$$ for any $$p \not = 2$$. We also prove the gradient estimate $$\Vert \nabla e^{-t\Delta }\Vert _{p-p} \le C t^{-\frac{1}{p}}$$, where $$\Delta $$ is the Laplace–Beltrami operator (acting on functions). Finally we discuss heat kernel bounds on forms and the Riesz transform on $$L^p$$ for $$p > 2$$.

Riesz transforms of the Hodge-de Rham Laplacian on Riemannian manifolds

Let $M$ be a complete non-compact Riemannian manifold satisfying the doubling volume property. Let $\overrightarrow{\Delta}$ be the Hodge-de Rham Laplacian acting on $1$-differential forms. According to the Bochner formula, $\overrightarrow{\Delta}=\nabla^*\nabla+R_+-R_-$ where $R_+$ and $R_-$ are respectively the positive and negative part of the Ricci curvature and $\nabla$ is the Levi-Civita connection. We study the boundedness of the Riesz transform $d^*(\overrightarrow{\Delta})^{-\frac{1}{2}}$ from $L^p(\Lambda^1T^*M)$ to $L^p(M)$ and of the Riesz transform $d(\overrightarrow{\Delta})^{-\frac{1}{2}}$ from $L^p(\Lambda^1T^*M)$ to $L^p(\Lambda^2T^*M)$. We prove that, if the heat kernel on functions $p_t(x,y)$ satisfies a Gaussian upper bound and if the negative part $R_-$ of the Ricci curvature is $\epsilon$-sub-critical for some $\epsilon\in[0,1)$, then $d^*(\overrightarrow{\Delta})^{-\frac{1}{2}}$ is bounded from $L^p(\Lambda^1T^*M)$ to $L^p(M)$ and $d(\overrightarrow{\Delta})^{-\frac{1}{2}}$ is bounded from $L^p(\Lambda^1T^*M)$ to $L^p(\Lambda^2T^* M)$ for $p\in(p_0',2]$ where $p_0>2$ depends on $\epsilon$ and on a constant appearing in the doubling volume property. A duality argument gives the boundedness of the Riesz transform $d(\Delta)^{-\frac{1}{2}}$ from $L^p(M)$ to $L^p(\Lambda^1T^*M)$ for $p\in [2,p_0)$ where $\Delta$ is the non-negative Laplace-Beltrami operator. We also give a condition on $R_-$ to be $\epsilon$-sub-critical under both analytic and geometric assumptions.

The Hodge–de Rham Laplacian and [formula omitted]-boundedness of Riesz transforms on non-compact manifolds

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform dΔ−12 on both Hardy spaces Hp and Lebesgue spaces Lp under two different conditions on the negative part R− of the Ricci curvature. First we prove that if R− is α-subcritical for some α∈[0,1), then the Riesz transform d∗Δ⃗−12 on differential 1-forms is bounded from the associated Hardy space HΔ⃗p(Λ1T∗M) to Lp(M) for all p∈[1,2]. As a consequence, dΔ−12 is bounded on Lp for all p∈(1,p0) where p0>2 depends on α and the constant appearing in the doubling property. Second, we prove that if∫01‖|R−|12v(⋅,t)1p1‖p1dtt+∫1∞‖|R−|12v(⋅,t)1p2‖p2dtt<∞, for some p1>2 and p2>3, then the Riesz transform dΔ−12 is bounded on Lp for all 1<p<p2. Furthermore, we study the boundedness of the Riesz transform of Schrödinger operators A=Δ+V on Lp for p>2 under conditions on R− and the potential V. We prove both positive and negative results on the boundedness of dA−12 on Lp.

The Hodge–de Rham Laplacian and Tachibana operator on a compact Riemannian manifold with curvature operator of definite sign

We give a comparative analysis of the spectral properties of the Hodge– de Rham and Tachibana operators on compact Riemannian manifolds whose curvature operator is bounded and has a definite sign. We find bounds for their spectra and estimate their multiplicities.

The spectral theory of the Yano rough Laplacian with some of its applications

J.H. Sampson has defined the Laplacian \(\triangle _\mathrm{sym}\) acting on the space of symmetric covariant tensors on Riemannian manifolds. This operator is an analogue of the well-known Hodge–de Rham Laplacian \(\triangle \) which acts on the space of skew-symmetric covariant tensors on Riemannian manifolds. In the present paper, we perform properties analysis of Sampson operator which acts on one-forms. We show that the Sampson operator is the Yano rough Laplacian. We also find the biggest lower bounds of spectra of the Yano and Hodge–de Rham operators and obtain estimates of their multiplicities for the space of one-forms on compact Riemannian manifolds with negative and positive Ricci curvatures, respectively.

On perturbative instability of Pope-Warner solutions on Sasaki-Einstein manifolds

Given a Sasaki-Einstein manifold, M 7, there is the $ \mathcal{N}=2 $ supersymmetric AdS 4 × M 7 Freund-Rubin solution of eleven-dimensional supergravity and the corresponding non-supersymmetric solutions: the perturbatively stable skew-whiffed solution, the perturbatively unstable Englert solution, and the Pope-Warner solution, which is known to be perturbatively unstable when M 7 is the seven-sphere or, more generally, a tri-Sasakian manifold. We show that similar perturbative instability of the Pope-Warner solution will arise for any regular Sasaki-Einstein manifold, M 7, admitting a basic, primitive, transverse (1,1)-eigenform of the Hodge-de Rham Laplacian with the eigenvalue in the range between $ 2\left( {9-4\sqrt{3}} \right) $ and $ 2\left( {9+4\sqrt{3}} \right) $ . Existence of such (1,1)-forms on all homogeneous Sasaki-Einstein manifolds can be shown explicitly using the Kähler quotient construction or the standard harmonic expansion. The latter shows that the instability arises from the coupling between the Pope-Warner background and Kaluza-Klein scalar modes that at the supersymmetric point lie in a long Z-vector supermultiplet. We also verify that the instability persists for the orbifolds of homogeneous Sasaki-Einstein manifolds that have been discussed recently.

A Generalization of a Levitin and Parnovski Universal Inequality for Eigenvalues

In this paper, we derive “universal” inequalities for the sums of eigenvalues of the Hodge de Rham Laplacian on Euclidean closed submanifolds and of eigenvalues of the Kohn Laplacian on the Heisenberg group. These inequalities generalize the Levitin–Parnovski inequality obtained for the sums of eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain.

The differential form spectrum of quaternionic hyperbolic spaces

By using harmonic analysis and representation theory, we determine explicitly the L 2 spectrum of the Hodge–de Rham Laplacian acting on quaternionic hyperbolic spaces and we show that the unique possible discrete eigenvalue and the lowest continuous eigenvalue can both be realized by some subspace of hypereffective differential forms. Similar results are obtained also for the Bochner Laplacian.

Dirac operators on Lagrangian submanifolds

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge–de Rham Laplacian provided the complex structure identifies the spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.

Multiplicative functional for the heat equation on manifolds with boundary

By the Weitzenbock formula relating the Hodge–de Rham Laplacian and the covariant Laplacian for differential forms on a Riemannian manifold, the heat equation for differential forms is naturally associated with a matrix-valued Feynman– Kac multiplicative functional determined by the curvature tensor. The case of a closed manifold (without boundary) is well known and will be briefly reviewed below. In constrast, the case of manifolds with boundary is not well known, and for good reasons. Because the absolute boundary condition on differential forms is Dirichlet in the normal direction and Neumann in the tangential directions, the associated multiplicative functional is discontinuous and much more difficult to handle. Ikeda and Watanabe [6; 7] have dealt with this situation by using an excursion theory (for reflecting Brownian motion) that seems to have been created especially for this problem. In this paper we suggest a different approach that is based on an idea of approximation due to Airault [1]. This construction has the advantage that a key property of the multiplicative functional (i.e., the attendant Ito’s formula for this functional) follows almost automatically from the approximate multiplicative functional without resorting to excursion theory, thus greatly simplifying this part of the theory; see Theorem 3.7. Before coming to another and more important raison d’etre for the present work, we briefly review some relevant facts for a closed manifold. LetM be a compact, closed Riemannian manifold and let α0 be a 1-form onM. Consider the following initial value problem:  ∂α ∂t = 1 2 α,

A Topological Criterion for the Existence of Half‐Bound States

The following theorem is proved: if (M4n+1,g) is a complete Riemannian manifold and Σ ⊂ M is an oriented hypersurface partitioning M and with non‐zero signature, then the spectrum of the Hodge–deRham Laplacian is [0,∞]. This result is obtained by a new Callias‐type index. This new formula links half‐bound harmonic forms (that is, nearly L2 but not in L2) with the signature of Σ.

Harmonic analysis for differential forms on complex hyperbolic spaces

We use representation theory for the semisimple Lie group G = SU( n, 1) to develop the L 2 harmonic analysis for differential forms on the complex hyperbolic space H n ( C ). In this setting, most of the basic notions and results known for functions are generalized: the abstract Plancherel theorem, the spectrum of the Hodge-de Rham Laplacian, the spherical function theory, the spherical Fourier transform and the Fourier transform. In addition, we calculate explicitly the Plancherel measure and estimate the decay at infinity of the heat kernel H t ( e).

Heat kernel for nonminimal operators on a Kähler manifold

The heat kernel expansion for a general nonminimal operator on the spaces C∞(Λk) and C∞(Λp,q) is studied. The coefficients of the heat kernel asymptotics for this operator are expressed in terms of the Seeley coefficients for the Hodge–de Rham Laplacian.

Antisymmetric tensor fields on spheres: Functional determinants and non-local counterterms

The Hodge–de Rham Laplacian on spheres acting on antisymmetric tensor fields is considered. Explicit expressions for the spectrum are derived in a quite direct way, confirming previous results. Associated functional determinants and the heat kernel expansion are evaluated. Using this method, new non-local counterterms in the quantum effective action are obtained, which can be expressed in terms of Betti numbers.