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We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators.…

The Novikov-Shubin invariants for a non-compact Riemannian manifold M can be defined in terms of the large time decay of the heat operator of the Laplacian on square integrable p-forms on M. For the (2n+1)-dimensional Heisenberg group H,…

We study first and second order conformal symmetries of the Yamabe Laplacian on a general pseudo-Riemannian manifold and of the Paneitz operator on Einstein spaces. We show that first order conformal symmetries of the Yamabe operator induce…

We study the deformability of the symmetric Einstein metrics on the spaces $\mathrm{SU}(n)/\mathrm{SO}(n)$ and $\mathrm{SU}(2n)/\mathrm{Sp}(n)$, thereby concluding the problem to second order for all irreducible symmetric spaces. The…

For odd dimensional Poincar\'e-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<\ndemi$) which are $C^m$ (with $m\in\nn$) on the conformal compactification $\bar{X}$ of $X$. This is infinite dimensional for…

A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces…

We give explicit formulas for conformally invariant operators with leading term an $m$-th power of Laplacian on the product of spheres with the natural pseudo-Riemannian product metric for all $m$.

In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition,…

We give a complete classification of conformally covariant differential operators between the spaces of differential $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the restriction of…

For any hypersurface of a Riemannian manifold, recent works introduced the notions of extrinsic conformal Laplacians and extrinsic Q-curvatures. Here we announce explicit formulas for the extrinsic Paneitz operators P_4 and the…

For a pseudo-Riemannian manifold $X$ and a totally geodesic hypersurface $Y$, we consider the problem of constructing and classifying all linear differential operators $\mathcal{E}^i(X) \to \mathcal{E}^j(Y)$ between the spaces of…

In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace and Cheng-Yau operators, on a bounded domain in a complete…

We show that on conformal manifolds of even dimension $n\geq 4$ there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian $\Delta^{k}$ for $k>n/2$. This shows that a…

We study a family of higher-derivative conformal operators $P_{2k}^{(2)}$ acting on transverse-traceless symmetric 2-tensors on generic Einstein spaces. They are a natural generalization of the well-known construction for scalars. We first…

INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…

Let $(M,g)$ be a pseudo-Riemannian manifold and $F_\lambda(M)$ the space of densities of degree $\lambda$ on $M$. We study the space $D^2_{\lambda,\mu}(M)$ of second-order differential operators from $F_\lambda(M)$ to $F_\mu(M)$. If $(M,g)$…

This paper deals with a certain class of second-order conformally invariant operators acting on functions taking values in particular (finite-dimensional) irreducible representations of the orthogonal group. These operators can be seen as a…

We give a complete classification of conformally covariant differential operators between the spaces of $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$. Moreover, we find explicit formul{\ae} for…

A topological version of four-dimensional (Euclidean) Einstein gravity which we propose regards anti-self-dual 2-forms and an anti-self-dual part of the frame connections as fundamental fields. The theory describes the moduli spaces of…

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators…