A Note on Lower Bounds Estimates for the Neumann Eigenvalues of Manifolds with Positive Ricci Curvature

We study new heat kernel estimates for the Neumann heat kernel on a compact manifold with positive Ricci curvature and convex boundary. As a consequence, we obtain lower bounds for the Neumann eigenvalues which are consistent with Weyl??s asymptotics.     

On Measure Contraction Property without Ricci Curvature Lower Bound

Measure contraction properties M C P (K, N) are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension N, then M C P (K, N) is equivalent to Ricci curvature bounded below by K. On the other hand, it was observed in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013) that there is a family of left invariant metrics on the three dimensional Heisenberg group for which the Ricci curvature is not bounded below. Though this family of metric spaces equipped with the Harr measure satisfy M C P (0,5). In this paper, we give sufficient conditions for a 2n+1 dimensional weakly Sasakian manifold to satisfy M C P (0, 2n + 3). This extends the above mentioned result on the Heisenberg group in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013).     

Lower Bounds for Morse Index of Constant Mean Curvature Tori

In the paper, three lower bounds are given for the Morse indexof a constant mean curvature torus in Euclidean3-space, in termsof its spectral genus g. The first two lower bounds grow linearlyin g and are stronger for smaller values of g, while the thirdgrows quadratically in g but is weaker for smaller values ofg. 2000 Mathematics Subject Classification 53A10, 53A35.     

Ricci Curvature and Fundamental Group

By refined volume estimates in terms of Ricci curvature, the two results due to J. Milnor (1968) are generalized.     

Functions of the Laplace Operator on Manifolds with Lower Ricci and Injectivity Bounds

Recent work of G. Mauceri, S. Meda, and M. Vallarino produces L ^p estimates on a natural class of functions of the Laplace–Beltrami operator on a Riemannian manifold M, under fairly weak geometrical hypotheses, namely lower bounds on its injectivity radius and Ricci tensor, but with an auxiliary decay hypothesis on the heat semigroup. We sharpen this result by removing the decay hypothesis.     

Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications

Consider a limit space ${(M_\alpha,g_\alpha,p_\alpha)\stackrel{GH}{\rightarrow} (Y,d_Y,p)}$ , where the ${M_\alpha^n}$ have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of Y at a point ${p\in Y}$ are known to be metric cones C(X), however they need not be unique. Let ${\overline\Omega_{Y,p}\subseteq\mathcal{M}_{GH}}$ be the closed subset of compact metric spaces X which arise as cross sections for the tangents cones of Y at p. In this paper we study the properties of ${\overline\Omega_{Y,p}}$ . In particular, we give necessary and sufficient conditions for an open smooth family ${\Omega\equiv (X,g_s)}$ of closed manifolds to satisfy ${\overline\Omega =\overline\Omega_{Y,p}}$ for some limit Y and point ${p\in Y}$ as above, where ${\overline\Omega}$ is the closure of Ω in the set of metric spaces equipped with the Gromov–Hausdorff topology. We use this characterization to construct examples which exhibit fundamentally new behaviors. The first application is to construct limit spaces (Y ⁿ , d _Y , p) with n ≥ 3 such that at p there exists for every 0 ≤ k ≤ n?2 a tangent cone at p of the form , where X ^n-k-1 is a smooth manifold not isometric to the standard sphere. In particular, this is the first example which shows that a stratification of a limit space Y based on the Euclidean behavior of tangent cones is not possible or even well defined. It is also the first example of a three dimensional limit space with nonunique tangent cones. The second application is to construct a limit space (Y ⁵ , d _Y , p), such that at p the tangent cones are not only not unique, but not homeomorphic. Specifically, some tangent cones are homeomorphic to cones over while others are homeomorphic to cones over .     

小Excess与具负下界Ricci曲率开流形的拓扑

In this paper,we study the relation between the excess of open manifolds and their topology by using the methods of comparison geometry.We prove that a complete open Riemmannian manifold with Ricci curvature negatively lower bounded is of finite topological type provided that the conjugate radius is bounded from below by a positive constant and its Excess is bounded by some function of its conjugate radius,which improves some results in [4].     

Invariant Measures and Asymptotic Gaussian Bounds for Normal Forms of Stochastic Climate Model

The systematic development of reduced low-dimensional stochastic climate models from observations or comprehensive high dimensional climate models is an important topic for atmospheric low-frequency variability, climate sensitivity, and improved extended range forecasting. Recently, techniques from applied mathematics have been utilized to systematically derive normal forms for reduced stochastic climate models for low-frequency variables. It was shown that dyad and multiplicative triad interactions combine with the climatological linear operator interactions to produce a normal form with both strong nonlinear cubic dissipation and Correlated Additive and Multiplicative (CAM) stochastic noise. The probability distribution functions (PDFs) of low frequency climate variables exhibit small but significant departure from Gaussianity but have asymptotic tails which decay at most like a Gaussian. Here, rigorous upper bounds with Gaussian decay are proved for the invariant measure of general normal form stochastic models. Asymptotic Gaussian lower bounds are also established under suitable hypotheses.     

Ricci Curvature and Bochner Formulas for Martingales

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 Bochner formula on M is related to lower bounds on Ricci curvature. Using this formalism, we obtain new characterizations of bounded Ricci curvature, new gradient estimates for martingales on path space, new Hessian estimates for martingales on path space, and streamlined proofs of the previous characterizations of bounded Ricci curvature.© 2018 Wiley Periodicals, Inc.     

Ricci Curvature Decay on Open Manifolds

The behaviour of the Ricci curvature along rays in a completeopen manifold is examined.     

Bounds for Maximal Functions Associated with Rotational Invariant Measures in High Dimensions

In recent articles (A. Criado in Proc. R. Soc. Edinb. Sect. A 140(3):541–552, 2010; Aldaz and Pérez Lázaro in Positivity 15:199–213, 2011) it was proved that when μ is a finite, radial measure in ? ⁿ with a bounded, radially decreasing density, the L ^p (μ) norm of the associated maximal operator M _μ grows to infinity with the dimension for a small range of values of p near 1. We prove that when μ is Lebesgue measure restricted to the unit ball and p<2, the L ^p operator norms of the maximal operator are unbounded in dimension, even when the action is restricted to radially decreasing functions. In spite of this, this maximal operator admits dimension-free L ^p bounds for every p>2, when restricted to radially decreasing functions. On the other hand, when μ is the Gaussian measure, the L ^p operator norms of the maximal operator grow to infinity with the dimension for any finite p>1, even in the subspace of radially decreasing functions.     

常曲率空间中具正Ricci曲率的子流形

设S~(n+p)(1)是一单位球面,M~n是浸入S~(n+p)(1)的具有非零平行平均曲率向量的n维紧致子流形.证明了当n≥4,p≥2时,如果M~n的Ricci曲率不小于(n-2)(1+H~2),则M~n是全脐的或者M~n的Ricci曲率等于(n-2)(1+H~2),进而M~n的几何分类被完全给出.     

Bakry–Emery Curvature Operator and Ricci Flow

In this paper, we introduce some techniques of Bakry–Emery curvature operator to Ricci flow and prove the evolution equation for the Bakry–Emery scalar curvature. As its application, we can easily derive the Perelman’s entropy functional monotonicity formula. We also discuss some gradient estimates of Ricci curvature and L ²– estimates of scalar curvature.Project partially supported by Yumiao Fund of Putian University.     

Complete Open Manifolds with Nonnegative Ricci Curvature

In this paper, we study complete open manifolds with nonnegative Ricci curvature and injectivity radius bounded from below. We find that this kind of manifolds are diffeomorphic to a Euclidean space when certain distance functions satisfy a reasonable condition.     

Curvature estimates for the Ricci flow I

In this paper we present several curvature estimates for solutions of the Ricci flow and the modified Ricci flow (including the volume normalized Ricci flow and the normalized Kähler-Ricci flow), which depend on the smallness of certain local \(L^{\frac{n}{2}}\) integrals of the norm of the Riemann curvature tensor |Rm|, where n denotes the dimension of themanifold. These local integrals are scaling invariant and very natural.     

Curvature estimates for the Ricci flow II

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kähler-Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of this norm. These results also serve as characterization of blow-up singularities.     

L-Curve and Curvature Bounds for Tikhonov Regularization

The L-curve is a popular aid for determining a suitable value of the regularization parameter when solving linear discrete ill-posed problems by Tikhonov regularization. However, the computational effort required to determine the L-curve and its curvature can be prohibitive for large-scale problems. Recently, inexpensively computable approximations of the L-curve and its curvature, referred to as the L-ribbon and the curvature-ribbon, respectively, were proposed for the case when the regularization operator is the identity matrix. This note discusses the computation and performance of the L- and curvature-ribbons when the regularization operator is an invertible matrix.