Related papers: The Lichnerowicz--Obata theorem for the Kohn Lapla…

The first variant of this article contained a fatal error. Therefore, we publish second version our paper. In the present paper, we prove that the curvature operator of the second kind of a Riemannian manifold is strictly positive if its…

Let $M$ be a finite volume oriented Riemannian manifold of dimension $n\geq 3$ and curvature in $[-b^2,-1]$, with thick-thin decomposition $M=M(thick)\cup M(thin)$. Denote by $\lambda_k(M(thick))$ the k-th eigenvalue for the Laplacian on…

We introduce a curvature-dimension condition for autonomous Lagrangians on weighted manifolds, which depends on the Euler-Lagrange dynamics on a single energy level. By generalizing Klartag's needle decomposition technique to the Lagrangian…

For a second order operator on a compact manifold satisfying the strong H\"ormander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class…

In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth…

In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold $M$ isometrically immersed into another Riemannian manifold $\bar M$ for arbitrary codimension. We first assume the pull back Weitzenb\"{o}ck…

We establish the $L^p$-boundedness of the local covariant Riesz transform for differential forms on manifold $M$ with bounded $\|Rm\|$. Let $\Delta_j$ be the Hodge Laplace operator on $j$-forms. For any $p \in (1, \infty)$ and…

We prove lower bound for the first closed or Neumann nonzero eigenvalue of the Laplacian on a compact quaternion-K\"ahler manifold in terms of dimension, diameter, and scalar curvature lower bound. It is derived as large time implication of…

In this paper, we obtain two Lichnerowicz type formulas for the Dirac-Witten operators. And we give the proof of Kastler-Kalau-Walze type theorems for the Dirac-Witten operators on 4-dimensional and 6- dimensional compact manifolds with…

In this paper, we establish a new eigenvalue estimate for the Kohn-Dirac operator on a compact CR manifold. The equality case of this estimate is characterized by the existence of a CR twistor spinor. We then classify CR manifolds carrying…

Given a strictly pseudoconvex CR manifold $M$ of dimension three and positive CR Yamabe class, and a positive smooth function $K:M\to\mathbf{R}$ verifying some mild and generic hypotheses, we prove the compactness of the set of solutions of…

In the paper, we prove the existence of a positive and essentially bounded solution to a Lichnerowicz equation in the Einstein-scalar field theory on a closed manifold with non-constant mean curvature. In particular, the non-constant mean…

This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4:…

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented pseudohermitian structures $\mathcal{P}_{+}$. We show that the functionals are continuous…

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues $\lambda_k$ of conformal sub-Riemannian metrics that are asymptotically sharp as $k\to…

We prove that optimal lower eigenvalue estimates of Zhong-Yang type as well as a Cheng-type upper bound for the first eigenvalue hold on closed manifolds assuming only a Kato condition on the negative part of the Ricci curvature. This…

In this paper we study bounds for the first eigenvalue of the Paneitz operator $P$ and its associated third-order boundary operator $B^3$ on four-manifolds. We restrict to orientable, simply connected, locally confomally flat manifolds that…

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces.…

We establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed K\"ahler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact…

We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang \cite{NiuZhang} for the…