arXiv research

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

B. Y. Chen established sharp inequalities between certain Riemannian invariants and the squared mean curvature for submanifolds in real space form as well as in complex space form. In this paper we generalize Chen inequalities for submanifolds of Bochner Kaehler manifolds. Moreover, we consider CR-warped product subman…

Introduces a universal Bochner formula for scalar curvature.

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…

At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes…

New proof of Gaffney's inequality for differential forms on manifolds with boundary.

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

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

Local logarithmic Brunn-Minkowski holds for zonoids.

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.

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric meas…

We develop the celebrated semigroup approach à la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenböck formula established by Sturm and the author. We show the $L^1$-gradient estimate on Finsler manifolds (under some additional assumptions in the n…

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

The study improves harmonic map theory for metric spaces with curvature bounds.

Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.

Study pseudoholomorphic maps using canonical connection.

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…

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the $Γ$-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev…

Derives sub-Riemannian Ricci curvature for various manifolds.

This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear an…

We use a modified Bochner technique to derive an inequality relating the nodal set of eigenspinors to eigenvalues of the Dirac operator on closed surfaces. In addition, we apply this technique to solutions of similar spinorial equations.

Unified approach to analysis on Ricci nonnegative manifolds and flows using optimal transport.

The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times.…

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

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…

Improved estimation of higher order integrals using shrinkage techniques.

New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.

We extend the range of $N$ to negative values in the $(K,N)$-convexity (in the sense of Erbar--Kuwada--Sturm), the weighted Ricci curvature $Ric_N$ and the curvature-dimension condition $CD(K,N)$. We generalize a number of results in the case of $N>0$ to this setting, including Bochner's inequality, the Brunn--Minkowsk…

We survey some $L^{p}$-vanishing results for solutions of Bochner or Simons type equations with refined Kato inequalities, under spectral assumptions on the relevant Schrödinger operators. New aspects are included in the picture. In particular, an abstract version of a structure theorem for stable minimal hypersurfaces…

We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold, where the L^2 norm of all second derivatives of the function involving horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian. Perturbation allows us to get a-priori bounds for solutions to sub-elliptic PDE in non-divergence…

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 …

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

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

We show that the configuration space over a manifold M inherits many curvature properties of the manifold. For instance, we show that a lower Ricci curvature bound on M implies for the configuration space a lower Ricci curvature bound in the sense of Lott-Sturm-Villani, the Bochner inequality, gradient estimates and Wa…

Proves Bochner's identity on graphs using a new auxiliary graph.

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…

We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold $\mathbb{M}$ without boundary, \begin{equation*} -Δ_g u_i = H_i(u_1,\cdots,u_m) \ \ \text{on} \ \ \mathbb{M}, \end{equation*} when $Δ_g$ stands for the Laplace-Beltrami operator, $u_i:\mathbb{M}\to \mathbb…

Study harmonic mappings and submanifolds using Bochner technique.

In this article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometri…

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

The Bochner tensor is the Kähler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)Kähler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms…

We apply the results from the article Cahen, Schwachhöfer: Special symplectic connections, to the case of Bochner-Kaehler metrics. We obtain a (local) classification of these based on the orbit types of the adjoint action in $su(n,1)$. The relation between Sasaki and Bochner-Kaehler metrics in cone and transveral metri…

We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-Kähler Vaisman manifolds.

New Bochner technique for foliations with non-negative Ricci curvature.

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

Study resolvents of Bochner Laplacians on compact manifolds.

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