Directional derivatives and generalized gradients on manifolds

For nonsmooth functions and differential forms on manifolds generalized directional derivatives, subgradients and Lie derivatives are introduced. Some rules for subgradients are given. Cartan’s formula and Stokes’ theorem are formulated for generalized subgradients and Lie derivatives     

The Szegö kernel on a class of non-compact CR manifolds of high codimension

We generalize Nagel’s formula for the Szegö kernel and use it to compute the Szegö kernel on a class of non-compact CR manifolds whose tangent space decomposes into one complex direction and several totally real directions. We also discuss the control metric on these manifolds and relate it to the size of the Szegö kernel.     

多孔介质方程在一般几何流演化下的梯度估计

本文研究了多孔介质方程在一般几何流下的梯度估计.通过Aroson和Bénilan对多孔介质方程的研究结果以及运用Li-Yau梯度估计的方法,获得了对多孔介质方程的正解对于Laplace算子以及drifing Laplace算子在一般几何流演化下的一些梯度估计,推广了Zhu Xiao-bao和Deng Yi-hua的结果...     

Spectral bounds for Dirac operators on open manifolds

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich’s estimate for manifolds with positive scalar curvature as well as the author’s estimate on surfaces.      

An approximation algorithm for quadratic dynamic systems based on N. Chomsky’s grammar for Taylor’s formula

Single-step methods for the approximate solution of the Cauchy problem for dynamic systems are discussed. It is shown that a numerical integration algorithm with a high degree of accuracy based on Taylor’s formula can be proposed in the case of quadratic systems. An explicit estimate is given for the remainder. The algorithm is based on N. Chomsky’s generative grammar for the language of terms of Taylor’s formula.     

A new entropy formula for the linear heat equation

We give a monotonicity entropy formula for the linear heat equation on complete manifolds with Ricci curvature bounded from below. As its applications, we get a differential Harnack inequality and a lower bound estimate about the heat kernel.     

New monotonicity formulas for Ricci curvature and applications. I

We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov?CHausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop?CGromov volume comparison theorem and Perelman??s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman??s monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li?CYau. In [CM4] one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds. Finally, there are obvious parallelisms between our monotonicity and the positive mass theorem of Schoen?CYau and Witten.     

A second variation formula for Perelman’s \mathcal W -functional along the modified Kähler-Ricci flow

We show a quite simple second variation formula for Perelman’s $\mathcal W $ -functional along the modified Kähler-Ricci flow over Fano manifolds.     

Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.     

General estimate of the first eigenvalue on manifolds

Ten sharp lower estimates of the first non-trivial eigenvalue of Laplacian on compact Riemannian manifolds are reviewed and compared. An improved variational formula, a general common estimate, and a new sharp one are added. The best lower estimates are now updated. The new estimates provide a global picture of what one can expect by our approach.     

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.     

加权流形上加权p-Laplace特征值问题的第一特征值下界估计

本文研究了加权流形上加权p-Laplacian特征值问题的第一特征值下界估计的问题.利用余面积公式、Cavalieri原理以及Federer-Fleming定理,获得了由Cheeger常数或等周常数确定的第一特征值的下界估计.     

Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds

Given a manifold \(M\) , we build two spherically symmetric model manifolds based on the maximum and the minimum of its curvatures. We then show that the first Dirichlet eigenvalue of the Laplace–Beltrami operator on a geodesic disk of the original manifold can be bounded from above and below by the first eigenvalue on geodesic disks with the same radius on the model manifolds. These results may be seen as extensions of Cheng’s eigenvalue comparison theorems, where the model constant curvature manifolds have been replaced by more general spherically symmetric manifolds. To prove this, we extend Rauch’s and Bishop’s comparison theorems to this setting.     

KOPPELMAN-LERAY FORMULA ON COMPLEX MANIFOLDS

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The entropy formula for linear heat equation

We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the manifold is isometric to the Euclidean space.     

Maslov’s canonical operator,Hörmander’s formula,and localization of the Berry-Balazs solution in the theory of wave beams

For three-dimensional Schrödinger equations, we study how to localize exact solutions represented as the product of an Airy function (Berry-Balazs solutions) and a Bessel function and known as Airy-Bessel beams in the paraxial approximation in optics. For this, we represent such solutions in the form of Maslov’s canonical operator acting on compactly supported functions on special Lagrangian manifolds. We then use a result due to Hörmander, which permits using the formula for the commutation of a pseudodifferential operator with Maslov’s canonical operator to “move” the compactly supported amplitudes outside the canonical operator and thus obtain effective formulas preserving the structure based on the Airy and Bessel functions. We discuss the influence of dispersion effects on the obtained solutions.     

Curvature vs. Almost Hermitian Structures

A Bochner-type formula for almost Hermitian manifolds is introduced. From this formula, one can find obstructions imposed by the curvature to the existence of certain almost Hermitian structures on compact manifolds.     

A family quantization formula for symplectic manifolds with boundary

This paper generalizes the family quantization formula of Zhang to the case of manifolds with boundary. As an application, Tian-Zhang’s analytic version of the Guillemin-Kalkman-Martin residue formula is generalized to the family case.     

The number of integer points on Vinogradov’s quadric

An asymptotic formula is obtained for the number of integer solutions of bounded height on Vinogradov’s quadric. Two leading terms are determined, and a strong estimate for the error term is given.     

THE DIFFERENTIAL INTEGRAL EQUATIONS ON SMOOTH CLOSED ORIENTABLE MANIFOLDS

1 IntroductionSillce tl1e limit value fOrlnula, viz. tl1e Plemelj fOrn1ula, of the Cauthe type integraJ withBochner-Martinelli kernel was proved in 1957[1], it has beell successfully used to the study Ofsingular i1ltegral equatious, solvi11g the 0b--equation, holomorphic extension, 0--closed exten-sion and C-R 111al1ifolds[2-51. Evideutly, the researcl1 of higher order singular integrals withBochuer-Martinelli kerllel itself also l1as important significallce. In 1952, J. Hadanmrd firstde…