arXiv research

We establish a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions, when the action group is unimodular.

New Roe C*-algebra for non-compact groupoids, leading to foliation Lichnerowicz vanishing.

Extends a theorem for non-compact spin manifolds with proper actions.

The study examines the geometry of Lichnerowicz Laplacian's kernel on various spaces.

Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds are reviewed. We present both in a unified approach using the representation of the Lie algebra of functions on itself by means of the hamiltonian vector fields. The use of the associated Lie algebroid allows to prove that the Lichnerowicz-Jacobi cohomolog…

Study invariant operators and vanishing theorems in CR geometry.

The paper extends Hodge-de Rham and Lichnérowicz Laplacians to double forms and proves vanishing theorems.

New rigidity results for tensors on non-compact manifolds with curvature conditions.

We construct Dirac operators on foliations by applying the Bismut-Lebeau analytic localization technique to the Connes fibration over a foliation. The Laplacian of the resulting Dirac operators has better lower bound than that obtained by using the usual adiabatic limit arguments on the original foliation. As a consequ…

The Euler class vanishes under certain curvature conditions.

We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], whi…

New findings on curvature and null spaces of Laplacians.

New eigenvalue bounds for 3-Sasaki metrics improve previous estimates.

Maximal index vanishes for certain spin manifolds with positive scalar curvature.

Extends results on metrics of positive scalar curvature.

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As…

We provide a complete list of two- and three-component Poisson structures of hydrodynamic type with degenerate metric, and study their homogeneous deformations. In the non-degenerate case any such deformation is trivial, that is, can be obtained via Miura transformation. We demonstrate that in the degenerate case this …

The paper studies rigidity results for harmonic forms on Kähler manifolds.

New proof shows solutions to Lichnerowicz equation exist.

The paper extends theorems on positive scalar curvature to foliated manifolds.

We construct a series of conformally invariant differential operators acting on weighted trace-free symmetric 2-tensors by a method similar to Graham-Jenne-Mason-Sparling's. For compact conformal manifolds of dimension even and greater than or equal to four with vanishing ambient obstruction tensor, one of these operat…

Proves existence and uniqueness of solutions to Lichnerowicz-type equations on manifolds.

We discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenboeck formula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac operators. For quaternionic Kaehler manifolds this leads to simple proofs of eigenvalue estima…

Paper finds a graph Steklov eigenvalue estimate with rigidity results.

Proves rigidity for eigenvalue estimate on three-manifolds.

New formula for Lichnerowicz Laplacian on homogeneous spaces.

Study on scalar curvature bounds and manifold topological complexity.

Study describes bifurcations and solutions of a specific equation on a particular manifold.

Proves existence of solution to Lichnerowicz equation on non-CMC manifolds.

Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.

Formula for Lichnerowicz Laplacian on invariant metrics, deducing stability of Einstein manifolds.

Proves existence of positive solutions to equations on manifolds with boundary.

The original version of the paper claimed to disprove the pseudo-Riemannian Lichnerowicz conjecture of D'Ambra and Gromov. However, the argument contains a crucial sign error in the lines following equation (8).

We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric.

Estimates eigenvalue for manifolds with specific forms under certain conditions.

In this paper we define a new cohomology of a smooth manifold called Lichnerowicz type cohomology attached to a function. Firstly, we study some basic properties of this cohomology as: a de Rham type isomorphism, dependence on the function, singular forms, relative cohomology, Mayer-Vietoris sequence, homotopy invarian…

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 compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere $S^n$. The spaces of symmetric eigentensors are explicitly given.

Study shows nonexistence of certain geometric structures in complex geometries.

In this article, we prove a Lichnerowicz estimate for a compact convex domain of a Kähler manifold whose Ricci curvature satisfies $\Ric \ge k$ for some constant $k>0$. When equality is achieved, the boundary of the domain is totally geodesic and there exists a nontrivial holomorphic vector field. We show that a ball o…

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of complex symmetric covariant tensor fields on the complex projective space $P^n(\comp)$. The spaces of symmetric eigentensors are explicitly given.

Counterexamples found for CR manifolds with mixed signature.

In this paper we generalize the basic Lichnerowicz cohomology on transversally locally conformally Kählerian foliations and we study its relation with basic Bott-Chern cohomology and $0$--th basic Dolbeault cohomology with values in the associated foliated weight bundle.

We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover).

In this paper we give a proof of Lichnerowicz Conjecture for compact simply connected manifolds which is intrinsic in the sense that it avoids the {\it Nice Embeddings} into eigen spaces of the Laplacian. Even if one wants to use these embeddings this paper gives a more streamlined proof.

Introduces a universal Bochner formula for scalar curvature.

We present a geometric construction of central extensions of covering groups of the group of volume preserving diffeomorphisms, integrating central extensions of the Lie algebra of divergence free vector fields defined by Lichnerowicz cocycles. Certain covering spaces of non-linear Grassmannians can be realized as preq…

In this paper we establish an analogue of the classical Lichnerowicz' theorem giving a sharp lower bound of the first non-zero eigenvalue of the sub-Laplacian on a compact seven-dimensional quaternionic contact manifold, assuming a lower bound of the qc-Ricci tensor, torsion tensor and its distinguished covariant deriv…