arXiv research

In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smoot…

In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

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

The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner type formulas for the Weyl tensor on a four dimensional Einstein manifold. In particular, we prove a second Bochner type formula which, formally, extends…

On Kahler manifolds with Ricci curvature bounded from below, we establish some theorems which are counterparts of some classical theorems in Riemannian geometry, for example, Bishop-Gromov's relative volume comparison, Bonnet-Meyers theorem, and Yau's gradient estimate for positive harmonic functions. The tool is a Boc…

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…

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…

The paper establishes eigenvalue inequalities for a specific operator on curved spaces.

Study on Riemannian Poisson warped product spaces and their properties.

Gradient and eigenvalue estimates for Kähler manifolds' canonical bundle.

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 …

We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manif…

Establish geometric and analytic constraints for Vafa-Witten equations on 4-manifolds, leading to volume bounds and new estimates for Yamabe constants.

We find some integral formulas of Simons and Bochner type and use them to study biharmonic and biconservative submanifolds in space forms. We obtain rigidity results that in the biharmonic case represent partial answers to two well-known conjectures on such submanifolds in spheres.

We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.

We consider almost $η$-Ricci solitons in $(LCS)_n$-manifolds satisfying certain curvature conditions. We provide a lower and an upper bound for the norm of the Ricci curvature in the gradient case, derive a Bochner-type formula for an almost $η$-Ricci soliton and state some consequences of it on an $(LCS)_n$-manifold.

Extends Bochner's theorem to include small positive Ricci curvature.

We consider almost quasi-Yamabe solitons in Riemannian manifolds, derive a Bochner-type formula in the gradient case and prove that under certain assumptions, the manifold is of constant scalar curvature. We also provide necessary and sufficient conditions for a gradient almost quasi-Yamabe soliton on the base manifold…

For an arbitrary Dirac-harmonic map $(φ,ψ)$ between compact oriented Riemannian surfaces, we shall study the zeros of $|ψ|$. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of $|ψ|$ and the genus of $M$ and $N$. On the basis, we could clarify all of nontrivial Dirac-har…

The paper finds a geometric lower bound for the first positive eigenvalue of the rough Laplacian on 1-forms.

Establishing Hom-versions of Bochner theorems in pseudo-Riemannian Hom-Lie algebras

Paper classifies Einstein-type manifolds with parallel Ricci tensor.

We provide a general Böchner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero rad…

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 …

We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study compact Hermitian-Weyl manifolds with non-negative symmetric part of the Ricci tens…

In this paper, we propose a property which is a natural generalization of Kazhdan's property $(T)$ and prove that many, but not all, groups with property $(T)$ also have this property. Let $\G$ be a finitely generated group. One definition of $\G$ having property $(T)$ is that $H^1(\G,π,\fh)=0$ where the coefficient mo…

In this article we apply a Bochner type formula to show that on a compact conformally flat riemannian manifold (or half-conformally flat in dimension 4) certain types of orthogonal almost-complex structures, if they exist, give the absolute minimum for the energy functional. We give a few examples when such minimizers …

Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions. Svensson [7] has shown that this formula simplifies to a Bochner type formula when we are dealing with Kähler manifolds and holomorphic (integrable) distributions. Here, …

We give the definition of $L^p$-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds…

We compute a Bochner type formula for static three-manifolds and deduce some applications in the case of positive scalar curvature. We also explain in details the known general construction of the (Riemannian) Einstein (n+1)-manifold associated to a maximal domain of a static n-manifold where the static potential is po…

We go further on the study of harmonicity for almost contact metric structures already initiated by Vergara-Diaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric st…

We prove that under certain conditions on the mean curvature and on the Kaehler angles, a compact submanifold M of real dimension 2n, immersed into a Kaehler-Einstein manifold N of complex dimension 2n, must be either a complex or a Lagrangian submanifold of N, or have constant Kaehler angle, depending on n=1, n=2, or …

A parallel lightlike vector field on a Lorentzian manifold $X$ naturally defines a foliation $\mathcal{F}$ of codimension one. If either all leaves of $\mathcal{F}$ are compact or $X$ itself is compact admitting a compact leaf and the (transverse) Ricci curvature is non-negative then a Bochner type argument implies tha…

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compac…

We study a new class of rank two sub-Riemannian manifolds encompassing Riemannian manifolds, CR manifolds with vanishing Webster-Tanaka torsion, orthonormal bundles over Riemannian manifolds, and graded nilpotent Lie groups of step two. These manifolds admit a canonical horizontal connection and a canonical sub-Laplaci…

Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \cite{DW…

Refines d'Alembertian for signed Lorentz distance functions in metric measure spacetimes.

The distributional category bounds manifold invariants and imposes constraints.

The paper explores geometric properties of Riemannian warped product maps and their curvature.

We prove a Bochner type vanishing theorem for compact complex manifolds $Y$ in Fujiki class $\mathcal C$, with vanishing first Chern class, that admit a cohomology class $[α] \in H^{1,1}(Y,\mathbb R)$ which is numerically effective (nef) and has positive self-intersection (meaning $\int_Y α^n \,>\, 0$, where $n\,=\,\di…

Localized curvature bounds ensure harmonic maps are constant.

Harmonic maps between specific metric spaces are studied with Lipschitz estimates and variational principles.

This research studies the smoothness of moduli spaces of self-dual contact instantons on Sasakian manifolds.

In this paper, we investigate critical maps of the horizontal energy functional $E_{H,\widetilde{H}}(f)$ for maps between two pseudo-Hermitian manifolds $(M^{2m+1},H(M),J,θ)$ and $(N^{2n+1},\widetilde{H}(N), \widetilde{J},\widetildeθ)$. These critical maps are referred to as $(H,\widetilde{H})$-harmonic maps. We derive…

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

New estimators outperform maximum likelihood without hyper-parameter estimation.

New estimator reduces kernel mean estimation error.