arXiv subjects

Roee Leder

Publications and source records attributed to Roee Leder.

5 recordsLinked to original sources

arXiv2025

Cohomology for linearized Ricci curvature

The Ricci curvature equations are a central subject of study in geometry. However, in the smooth real case, their linear analysis is often confined to settings in which the background metric is Einstein. In this paper, we establish solvability and uniqueness conditions for the linearized problem on any compact Riemannian manifold with boundary. These conditions are formulated in terms of the cohomology of a canonical cochain complex, constructed by means of a generalized Hodge theory based on pseudodifferential methods. An important element of the theory is that it allows the incorporation of tensorial error terms, arising from linearized metric-dependent sources or from connections on the manifold of metrics. Using Bochner technique, we prove vanishing theorems for the cohomology under geometric assumptions on the boundary and error term, without imposing further interior restrictions.

math.DG↗

arXiv2025

Hodge Theory for Linearized Boundary-Value Problems on General Geometric Structures

We develop a framework that systematically casts solvability and uniqueness conditions for linearized overdetermined boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the underlying geometric structure and provides tools to study the resulting cohomology explicitly. To achieve this generality, we develop the notion of an elliptic pre-complex, which generalizes the machineries of Hodge theory to sequences of Douglis--Nirenberg systems in the pseudodifferential calculus of boundary-value problems that interact through Green's formulae, the notion of overdetermined ellipticity, and a condition we call the order-reduction property, which relaxes the rigid requirement that the sequence form a cochain complex. This property typically arises from linearized symmetries and constraints, as we demonstrate through several geometric examples that have long resisted analysis, including exterior covariant derivatives, the Killing and Hessian equations, and the Riemann curvature equations.

math.DG↗

arXiv2023

Elliptic Pre-Complexes, Hodge-like Decompositions and Overdetermined Boundary-Value Problems

We solve a problem posed by Calabi more than 60 years ago, known as the Saint-Venant compatibility problem: Given a compact Riemannian manifold, generally with boundary, find a compatibility operator for Lie derivatives of the metric tensor. This problem is related to other compatibility problems in mathematical physics, and to their inherent gauge freedom. To this end, we develop a framework generalizing the theory of elliptic complexes for sequences of linear differential operators $(A_{\bullet})$ between sections of vector bundles. We call such a sequence an elliptic pre-complex if the operators satisfy overdetermined ellipticity conditions, and the order of $A_{k+1}A_k$ does not exceed the order of $A_k$. We show that every elliptic pre-complex $(A_{\bullet})$ can be "corrected" into a complex $(\mathcal{A}_{\bullet})$ of pseudodifferential operators, where $\mathcal{A}_k - A_k$ is a zero-order correction within this class. The induced complex $(\mathcal{A}_{\bullet})$ yields Hodge-like decompositions, which in turn lead to explicit integrability conditions for overdetermined boundary-value problems, with uniqueness and gauge freedom clauses. We apply the theory on elliptic pre-complexes of exterior covariant derivatives of vector-valued forms and double forms satisfying generalized algebraic Bianchi identities, thus resolving a set of compatibility and gauge problems, among which one is the Saint-Venant problem.

math.AP↗

arXiv2021

On Saint-Venant compatibility and stress potentials in manifolds with boundary and constant sectional curvature

We address three related problems in the theory of elasticity, formulated in the framework of double forms: the Saint-Venant compatibility condition, the existence and uniqueness of solutions for equations arising in incompatible elasticity, and the existence of stress potentials. The scope of this work is for manifolds with boundary of arbitrary dimension, having constant sectional curvature. The central analytical machinery is the regular ellipticity of a boundary-value problem for a bilaplacian operator, and its consequences, which were developed in [KL21]. One of the novelties of this work is that stress potentials can be used in non-Euclidean geometries, and that the gauge freedom can be exploited to obtain a generalization for the biharmonic equation for the stress potential in dimensions greater than two.

math.AP↗

arXiv2021

Double forms: Regular elliptic bilaplacian operators

Double forms are sections of the vector bundles $\Lambda^{k}T^*\mathcal{M}\otimes \Lambda^{m}T^*\mathcal{M}$, where in this work $(\mathcal{M},\mathfrak{g})$ is a compact Riemannian manifold with boundary. We study graded second-order differential operators on double forms, which are used in physical applications. A Combination of these operators yields a fourth-order operator, which we call a double bilaplacian. We establish the regular ellipticity of the double bilaplacian for several sets of boundary conditions. Under additional conditions, we obtain a Hodge-like decomposition for double forms, whose components are images of the second-order operators, along with a biharmonic element. This analysis lays foundations for resolving several topics in incompatible elasticity, most prominently the existence of stress potentials and Saint-Venant compatibility.

math.AP↗