arXiv research

Introduces a universal Bochner formula for scalar curvature.

Study Bochner formula on metric measure spaces for vanishing Betti numbers.

The paper derives inequalities and formulas for generalized Ricci flow.

We generalize the classical Bochner formula for the heat flow on evolving manifolds $(M,g_{t})_{t \in [0,T]}$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $P\mathcal{M}$ of space-time $\mathcal{M} = M \times [0,T]$. Our new Bochner formula and the inequalities that follow from it a…

We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical…

In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…

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…

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.

Einstein's non-symmetric geometry uses Bochner's technique to prove decomposition and vanishing results.

New formulas with quadratic curvature terms on Kähler manifolds for Hodge number estimates.

We prove the Bochner-Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li-Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry-Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.

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, …

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra, which induce not only identities for higher Casimir elements but also all Bochne…

We introduce the study of nonlinear harmonic forms. These are forms which minimize the $L_2$ energy in a cohomology class subject to a nonlinear constraint. In this note, we include only motivations and the most basic existence results. We also introduce a variant of the Bochner formula suitable for probing the structu…

The paper studies eigenvalues in gaps of the essential spectrum of a Bochner-Schrödinger operator.

New derivation of Type IIA flow metrics.

A Kahler metric is said to be Bochner-Kahler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kahler metrics in complex dimension n has real …

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT($κ$) spaces for $κ\in\{0,1\}$. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT($κ$) spaces. We compute a target variation formula that captures the curvature bound in…

In this short paper, we re-derive the Bochner formula for the Laplacian by considering local variations of volume. The derivation is rooted in the fact that the Laplacian of a function measures the volume variation along the flow of the gradient vector of the function. Possible extensions of this approach/technique are…

Study the Bochner-Schrödinger operator's trace in semiclassical limit.

We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and Kähler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we…

Study pseudoholomorphic maps using canonical connection.

We consider the renormalized Bochner Laplacian acting on tensor powers of a positive line bundle on a compact symplectic manifold. We derive an explicit local formula for the spectral density function in terms of coefficients of the Riemannian metric and symplectic form.

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eels-Sampson…

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…

We give a generalization of the Penrose transform on Hermitian manifolds with metrics locally conformally equivalent to Bochner-Kähler metrics. We also give an explicit formula for the inverse transform. This paper is a generalization of "The Twistor correspondence of the Dolbeault complex over $\C^n$" (dg-ga/9501004) …

Study of Riemann solitons and $η$-hyperbolic Ricci solitons on Bochner-flat Lorentzian Kähler spacetime manifolds.

The paper extends Bochner's technique to singular distributions on manifolds.

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.

The report presents the theory of harmonic maps from Kähler manifolds.

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 compute the first two equivariant heat kernel coefficients of the Bochner Laplacian on differential forms. The first two equivariant heat kernel coefficients of the Bochner Laplacian with torsion are also given. We also study the equivariant heat kernel coefficients of nonmininmal operators on differe…

We study the dynamical behaviors of degenerate stochastic differential equations (SDEs). We select an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conduct the Lyapunov exponential convergence analysis of degenerate SDEs. We derive the convergence rate cond…

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 derive some $\partial\overline{\partial}$-Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a comapct Hermitian manif…

Study on almost Riemann solitons with gradient or torse-forming vector fields.

The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge's theory, Bochner-Weitzenbock and Simon's type formulas. Moreover, a few global and local theorem…

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.

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…

The study develops inequalities for Riemannian foliations without bundle-like assumptions.

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…

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

Derives sub-Riemannian Ricci curvature for various manifolds.

In this paper, we consider a Riemannian manifold (M, g) endowed with a Riemannian flow and we study the curvature term in the Bochner-Weitzenb{ö}ck formula of the basic Laplacian on M. We prove that this term splits into two parts. The first part depends mainly on the curvature operator of the underlying manifold M and…

New positivity condition for Hermitian manifold curvature.

We show that the group of smooth isometries of a complemented sub-Riemannian manifold form a Lie group and establish dimension estimates based on the torsion of the canonical connection. We explore the interaction of curvature and the structure of isometries and Killing fields and derive a Bochner formula for Killing f…

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq k$ using only curvature terms in the Weitzenböck formulae for symmetric $p$-tensors. By introducing a symmetric analogue of the Kulkarni-Nomizu product, we provide a simple formula for such curvature terms. We also gi…

The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.