arXiv research

Introduces a universal Bochner formula for scalar curvature.

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

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 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.

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

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

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

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

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.

Study invariant operators and vanishing theorems in CR geometry.

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalize…

Study stability of Einstein metrics on homogeneous spaces.

New formula for Lichnerowicz Laplacian on homogeneous spaces.

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…

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 …

Contact Riemannian manifolds, with not necessarily integrable complex structures, are the generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection on such a manifold plays the role of Tanaka-Webster connection in the pseudohermitian case. We prove the contact Riemannian version of…

In this paper, we describe the group SpinT (n) and give some properties of this group. We construct SpinT spinor bundle S by means of the spinor representation of the group SpinT (n) and define covariant derivative operator and Dirac operator on S. Finally, Schrodinger-Lichnerowicz-type formula is derived by using thes…

Paper finds a graph Steklov eigenvalue estimate with rigidity results.

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,…

Proves rigidity for eigenvalue estimate on three-manifolds.

Study shows instability of Kähler Ricci solitons and stability of orbifold singularities.

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.

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…

New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the…

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

Under appropriate spectral assumptions we prove two existence results for positive solutions of Lichnerowicz-type equations on complete manifolds. We also give a priori bounds and a comparison result that immediately yields uniqueness for certain classes of solutions. No curvature assumptions are involved in our analys…

In this paper, we establish an intrinsic Gauss-Bonnet-Chern formula for Finsler manifolds by using the Mathai-Quillen's superconnection formalism, in which no extra vector field is involved. Furthermore, we prove a more general Lichnerowicz formula in this direction through a geometric localization procedure.

In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman's $ν$-entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary condition for linearly …

The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant $r$-mean curvature. We prove a generalized Bochner-type formula for such a kind of operators and as applications we obtain some sharp estimat…

Study shows nonexistence of certain geometric structures in complex geometries.

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 …

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.

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

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…

Researchers study metrics with maximal Ricci curvature on homogeneous spaces.

We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\leq p \leq n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\leq p<n/2$.

Derives scalar curvature formula in generalized Kähler geometry.

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…

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 main technical result of the paper is a Bochner type formula for the sub-laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the $Sp(n)Sp(1)$ components of the …

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

In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -Δu=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$, $B(x)\geq0$ are given smooth functions. Our analysis is based on the global existence of positive solution…

In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.

We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on 3-manifolds with positive Ricci curvature. In our estimate we make no assumption on the …

Consider a Riemannian spin manifold $(M^{n}, g)$ $(n\geq 3)$ endowed with a non-trivial 3-form $T\inΛ^{3}T^{*}M$, such that $\nabla^{c}T=0$, where $\nabla^{c}:=\nabla^{g}+\frac{1}{2}T$ is the metric connection with skew-torsion $T$. In this note we introduce a generalized $\frac{1}{2}$-Ricci type formula for the spinor…