Lower bounds for eigenvalues of the Dirac-Witten operator

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator of compact(with or without boundary) spacelike hypersurfaces of Lorentian manifold satisfying certain conditions,just in terms of the mean curvature and the scalar curvature and the spinor energy-momentum tensor. In the limiting case,the spacelike hypersurface is either maximal and Einstein manifold with positive scalar curvature or Ricci-flat manifold with nonzero constant mean curvature.     

Hypersurfaces with constant scalar curvature and constant mean curvature

Non-spherical hypersurfaces inE ⁴ with non-zero constant mean curvature and constant scalar curvature are the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.     

局部对称Lorentz空间中的类空超曲面

研究了局部对称Lorentz空间中具有常平均曲率或常数量曲率的类空超曲面.利用丘成桐的广义极大值原理和自伴随算子得到了两个重要的内蕴刚性定理,其分别推广了欧阳崇祯和刘新民的主要结果.     

Spectral Properties of Constant Mean Curvature Submanifolds in Hyperbolic Space

In this work we determine the essential spectrum of the stability operator of a submanifold of the hyperbolic space with constant mean curvature h < 1 and finite total curvature. In some particular cases, we also give a bound on the number of eigenvalues which are below the essential spectrum.     

Almost nonnegative curvature operator and cohomology rings

We give a survey of results on the construction of and obstructions to metrics of almost nonnegative curvature operator on closed manifolds and results on the cohomology rings of closed, simply-connected manifolds with a lower curvature and upper diameter bound. The latter is motivated by a question of Grove whether these condition imply finiteness of rational homotopy types. This question has answers by F. Fang–X. Rong, B. Totaro and recently A. Dessai and the present author.     

Riemannian Manifold in Which the Skew-Symmetric Curvature Operator has Pointwise Constant Eigenvalues

We study manifolds where the natural skew-symmetric curvature operator has pointwise constant eigenvalues. We give a local classification (up to isometry) of such manifolds in dimension 4. In dimension 3, we describe such manifolds up to a classification of three - dimensional Riemannian manifolds with principal Ricci curvatures r₁ = r₂ = 0, r₃- arbitrary. We give examples of such manifolds in all dimensions which do not have constant sectional curvature; these manifolds are not pointwise Osserman manifolds in general.     

Sharp lower bounds for the first eigenvalues of the bi-drifting Laplacian

In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth metric measure space) and weighted Ricci curvature bounded inferiorly.     

空间形式中具常数量曲率的子流形的内蕴刚性

Under the assumption that the normalized mean curvature vector is parallel in the normal bundle, by using the generalized ChengYau＇s self-adjoint differential operator, here we obtain some rigidity results for compact submanifolds with constant scalar curvature and higher codimension in the space forms.     

局部对称空间中具有常数量曲率的超曲面

利用自伴算子研究局部对称空间中具有常数量曲率的紧致超曲面,得到了这类超曲面中的某些刚性定理,推广了已有的结果.     

CR Geometry in 3-D

CR geometry studies the boundary of pseudo-convex manifolds.By concentrating on a choice of a contact form,the local geometry bears strong resemblence to conformal geometry.This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three.While the sign of this operator is important in the embedding problem,the kernel of this operator is also closely connected with the stability of CR structures.The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed.The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem.The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension,and it is closely connected with the pluriharmonic functions.The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms.Finally,an isoperimetric constant determined by the Q-prime curvature integral is discussed.     

First stability eigenvalue characterization of CMC Hopf tori into Riemannian Killing submersions

We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature orientable surfaces immersed in a Riemannian Killing submersion. As a consequence, the strong stability of such surfaces is studied. We also characterize constant mean curvature Hopf tori as the only ones attaining the bound in certain cases.     

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.     

On the complete linear Weingarten spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

We deal with complete linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form, having two distinct principal curvatures. In this setting, we show that such a spacelike hypersurface must be isometric to a certain isoparametric hypersurface of the ambient space, under suitable restrictions on the values of the mean curvature and of the norm of the traceless part of its second fundamental form. Our approach is based on the use of a Simons type formula related to an appropriated Cheng–Yau modified operator jointly with some generalized maximum principles.     

Ground state solution for a problem with mean curvature operator in Minkowski space

In this paper we prove the existence of a radial ground state solution for a quasilinear problem involving the mean curvature operator in Minkowski space.     

Sharp upper bounds for \lambda _1^{L_r } of immersed hypersurfaces and their stability in space forms

In this paper we derive new sharp upper bounds for the first positive eigenvalue λ1^Lr of the linearized operator of the higher order mean curvature of a closed hypersurface immersed into a Riemannian space form R^（n＋l）（c） （c 〉 0）. Our bounds are extrinsic in the sense that they are given in terms of the higher order mean curvatures. Under the assumption Hr＋2 〉 0, by establishing two valuable integral formulas, we obtain unified sharp upper bounds of λ1^Lr. We also give an estimation of the upper bounds of the first eigenvalue of a SchrSdinger-type operator, by which we prove those hypersurfaces with positive constant Hr＋l in any space forms are stable if and only if they are geodesic spheres, thus generalizing the previous result obtained only in the case c≤0.     

Riemannian Manifolds in Which Certain Curvature Operator Has Constant Eigenvalues along Each Circle

Riemannian manifolds for which a natural curvature operator has constant eigenvalues on circles are studied. A local classification in dimensions two and three is given. In the 3-dimensional case one gets all locally symmetric spaces and all Riemannian manifolds with the constant principal Ricci curvatures r ₁ = r ₂ = 0, r ₃= 0 , which are not locally homogeneous, in general.     

Horizontal Connection and Horizontal Mean Curvature in Carnot Groups

In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction （projection） of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and （intrinsic） regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator.     

Obstructions to positive curvature and symmetry

We discuss new obstructions to positive sectional curvature and symmetry. The main result asserts that the index of the Dirac operator twisted with the tangent bundle vanishes on a 2-connected manifold of dimension ≠8 if the manifold admits a metric of positive sectional curvature and isometric effective S¹-action. The proof relies on the rigidity theorem for elliptic genera and properties of totally geodesic submanifolds.     

Properties of Berwald scalar curvature

We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature. As a consequence, Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds. For (α,β)-metrics on manifold of dimension greater than 2, if the mean Landsberg curvature and the Berwald scalar curvature both vanish, then the Berwald curvature also vanishes.     

The first Dirac eigenvalues on manifolds with positive scalar curvature

We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich's eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.