arXiv research

Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general inequality for them. As an application, by using this inequality, we study eigenvalues…

Study eigenvalues and eigenfunctions of fourth-order operators in annuli, proving optimal estimates and non-radiality.

In this note we review some results regarding higher order elliptic differential operators on manifolds without boundary.

New framework uses elliptic operators to study projective maps.

We prove the compactness of solutions to general fourth order elliptic equations which are L^1-perturbations of the Q-curvature equation on compact Riemannian 4-maniods. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by product, we …

Using a method developped in [1] and [2], we prove the existence of weak non trivial solutions to fourth order elliptic equations with singularities and with critical Sobolev growth.

The paper provides estimates for eigenvalues of elliptic differential problems.

This paper deals with the existence of solutions to a class of fourth order nonlinear elliptic equations. The technique used relies on critical points theory. The solutions appeared as critical points of a functional restricted to a suitable manifold.In the case of constant coefficients we obtain the existence of tree …

We study the conformal plate buckling equation (Laplace--Beltrami)^2 u =1, where the L-B operator is for the metric g = e^{2u}g_0, with $g_0$ the standard Euclidean metric on R^2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order, Delta u +K_g e^(2u)=0, D…

Using the method of Nehari manifold, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singulatities and with critical Sobolev growth.

In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constan…

In this paper we prescribe a fourth order conformal invariant on the standard $n-$sphere, with $n\geq5$, and study the related fourth order elliptic equation. We first find some existence results in the perturbative case. After some blow up analysis we build a homotopy to pass from the perturbative case to the non-pert…

Smoothness of Hamiltonian stationary submanifolds in symplectic manifolds proven.

In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of ba…

The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kaehler (cscK) metrics, when the cone angle is less than $π$. We introduce a new Hölder space called $\cC^{4,\a,\b}$ to study the regularities of this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$ conic cscK metri…

In present paper, the equivalence problem for fourth order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem i…

In this paper, we study some fourth order singular critical equations of Lichnerowicz type involving the Paneitz-Branson operator, and we prove existence and non existence results under given assumptions.

We study compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds of dimension $\ge 5$. If the $Q$-curvature equals $-1$, we prove that all solutions are universally bounded. If the $Q$-curvature is $1$, assuming that Paneitz operator's kernel is …

The paper examines scalar fourth-order linear differential operators and their invariants.

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a priori estimates in dimension 5 and 6. On the five dimensionl sphere these a prio…

In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary there exists a metric conformal to g with constant T-curvature, zero Q-curvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth order …

In this paper we continue our study of the fourth order transgression on hyperähler manifolds introduced in the previous paper. We give a local construction for the fourth-order transgression of the Chern character form of an arbitrary vector bundle supplied with a self-dual connection on a four dimensional hyperkähler…

We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. Thi…

Proves a higher-order positive energy theorem for stationary solutions in fourth-order gravity.

For the dual operator $s_g'^*$ of the linearization $s_g'$ of the scalar curvature function, it is well-known that if $\ker s_g'^*\neq 0$, then $s_g$ is a non-negative constant. In particular, if the Ricci curvature is not flat, then ${s_g}/(n-1)$ is an eigenvalue of the Laplacian of the metric $g$. In this work, some…

In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities chang…

We propose a generalization of the Hodge $dd_c$-lemma to the case of hyperkähler manifolds. As an application of this result we derive the global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperkähler manifolds. At the second part of the paper we…

We discuss some open problems and recent progress related to the 4th order Paneitz operator and Q curvature in dimensions other than 4.

Stability of branched immersions with energy constraints.

For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kern…

This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using $KK$-theory.

Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encomp…

New operators and curvatures derived from embedded manifolds.

A nonstandard invariant fourth order operator acting on functions on a manifold equipped with an almost Grassmannian structure with an arbitrary trorsion is found by means of the curved translation principle. This operator can be viewed as a Grassmannian analogue of the Paneitz operator well known from conformal geomet…

In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an $L^\infty$ Kähler metric. The main result is to show that such a weak solution (with uniform $L^\infty$ bound…

Extends a theorem for first-order elliptic operators on manifolds.

Introduces a new elliptic operator with positive eigenvalue.

New Witten rigidity theorems for elliptic genus in various dimensions.

An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of …

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the …

Let $(\mathbf{M}^{3},J,θ_{0})$ be a closed pseudohermitian 3-manifold. Suppose the associated torsion vanishes and the associated $Q$-curvature has no kernel part with respect to the associated Paneitz operator. On such a background pseudohermitian 3-manifold, we study the change of the contact form according to a cert…

In this note the fractional analytic index, for a projective elliptic operator associated to an Azumaya bundle, of DG/0402329 is related to the equivariant index of Atiyah and Singer for an associated transversally elliptic operator.

New calculus solves boundary value problems for elliptic operators.

The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.

We introduce a fourth order CR invariant operator on pluriharmonic functions on a three-dimensional CR manifold, generalizing to the abstract setting the operator discovered by Branson, Fontana and Morpurgo. For a distinguished class of contact forms, all of which have vanishing Hirachi-$Q$ curvature, these operators d…

The paper studies elliptic operators on manifolds with boundary.