arXiv research

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

We prove a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions. This extends a previous result of Ziran Liu who proves it for the case where the acting group is unimodular.

Proves rigidity for eigenvalue estimate on three-manifolds.

The paper proves formulas and theorems for J-Witten deformation on specific manifolds.

The paper proves formulas and theorems for sub-signature operators on manifolds with or without boundaries.

The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.

The paper proves formulas and theorems for Dirac-Witten operators on manifolds with or without boundaries.

The paper proves a theorem for a twisted Dirac operator on specific manifolds.

The paper proves a theorem for a twisted Dirac operator on specific manifolds.

There are very few general theorems on the kernel of the well-known Lichnerowicz Laplacian. In the present article we consider the geometry of the kernel of this operator restricted to covariant (not necessarily symmetric or skew-symmetric) tensors. Our approach is based on the analytical method, due to Bochner, of pro…

The paper proves formulas and theorems for specific operators on manifolds.

We exhibit two three-parameter families of locally conformal symplectic forms on the solvmanifold $M_{n,k}$ considered in [1], and show, using the Hodge-de Rham theory for the Lichnerowicz cohomology that that they are not $d_ω$ exact, i.e. their Lichnerowicz classes are non-trivial (Theorem 1). This has several import…

Study invariant operators and vanishing theorems in CR geometry.

The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.

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…

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the …

Study eigenvalues of ellipsoids near a sphere, comparing to sphere's.

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…

A classical result in differential geometry due to Lichnerowicz [8] is concerned with the decomposition of the square of Dirac operators defined by Clifford connections on a Clifford module ${\cal E}$\ over a Riemannian manifold $M$. Recently, this formula has been generalized to arbitrary Dirac operators [2]. In this …

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

We obtain a Bochner type formula and an estimate from below on the spectrum of the sublaplacian of a compact strictly pseudoconvex CR manifold.

We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the…

The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.

We introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid e…

In this paper, we give two Lichnerowicz type formulas for modified Novikov operators. We prove KastlerKalau-Walze type theorems for modified Novikov operators on compact manifolds with (resp.without) boundary. We also compute the spectral action for Witten deformation on 4-dimensional compact manifolds.

The aim of this paper is to prove a normal form Theorem for Dirac-Jacobi bundles using the recent techniques from Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As an application we provide a alt…

Study new Einstein-like metrics and their properties.

Introduces new equations linking Kähler-Einstein and Hermitian-Yang-Mills theories.

In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we prove isomorphisms between twisted Morse homology, Steenrod's CW-homology with loca…

Given a complete, smooth metric measure space $(M,g,e^{-f}dv)$ with the Bakry-Émery Ricci curvature bounded from below, various gradient estimates for solutions of the following general $f$-heat equations $u_t=Δ_f u+au\log u+bu +Au^p+Bu^{-q}$ and \[ u_t=Δ_f u+Ae^{pu}+Be^{-pu}+D \] are studied. As by-product, we obt…

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

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…

We prove the following generalization of the classical Lichnerowicz vanishing theorem: if $F$ is an oriented flat vector bundle over a closed spin manifold $M$ such that $TM$ carries a metric of positive scalar curvature, then $<\widehat A(TM)e(F),[M]>=0$, where $e(F)$ is the Euler class of $F$.

Paper studies eigenvalue bounds for complex curves on Kähler surfaces.

Paper finds a graph Steklov eigenvalue estimate with rigidity results.

In this paper, we give two Lichnerowicz type formulas for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection. We also prove two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on 4-dimensional manifolds with (resp. without) boun…

Inspired by the Lichnerowicz-Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants $\{Ω_k(g)\}$ for closed Riemannian manifolds. The value of $Ω_k(g)$ delicately reflects the spherical part of the manifold. Indeed, $Ω_1(g)$ and $Ω_2(g)$ characterize the standard sphere.

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor,…

Let M be a riemannian manifold. The existence of a spin structure on M, enables to study the topology of M. The obstruction to the existence of the spin structure is given by the second Stiefel-Whitney class. This class is the classifying cocycle of a gerbe. One may expect that the study of this gerbe may have topologi…

We prove a lower bound for the first eigenvalue of the sub-Laplacian on sub-Riemannian manifolds with transverse symmetries. When the manifold is of H-type, we obtain a corresponding rigidity result: If the optimal lower bound for the first eigenvalue is reached, then the manifold is equivalent to a 1 or a 3-Sasakian s…

We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can pe…

New formula for Lichnerowicz Laplacian on homogeneous spaces.

In this paper we extend to the difference case the notion of Poisson-Lichnerowicz cohomology, an object encapsulating the building blocks for the theory of deformations of Hamiltonian operators. A local scalar difference Hamiltonian operator is a polynomial in the shift operator and its inverse, with coefficients in th…

We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the fa…

We provide a complete study of existence and uniqueness of solutions to the Lichnerowicz equation in general relativity with arbitrary mean curvature.

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