紧致流形上Lichnerowicz方程的梯度估计(英文)

Let(M,g) be a smooth compact Riemannian manifold of dimension n.Denote△f=△-▽f.▽ the weighted Laplacian operator,where f is a smooth real valued function on M.When N is finite and the N-Bakry-Emery Ricci tensor is bounded from below by a constant,we establish local gradient estimates for positive solutions of the following simple Lichnerowicz equation△fu+cu~(-α)=0 on a compact Riemannian manifold,where α is a positive constant and c is a smooth function.     

Projective vector fields on Finsler manifolds

In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold with non-positive flag curvature and the non-existence of projective vector fields on the compact Finsler manifold with negative flag curvature. Furthermore, we deduce some expectable, but non-trivial relationships between geometric vector fields such as projective, affine, conformal, homothetic and Killing vector fields on a Finsler manifold.     

ASYMPTOTIC GROWTH BOUNDS FOR THE VLASOV-POISSON SYSTEM WITH RADIATION DAMPING

We consider asymptotic behaviors of the Vlasov-Poisson system with radiation damping in three space dimensions.For any smooth solution with compact support,we prove a sub-linear growth estimate of its velocity support.As a consequence,we derive some new estimates of the charge densities and the electrostatic field in this situation.     

Boundability of Open Manifolds

An open manifold in question is assumed to bea noncompact manifold with compact boundary (probably empty). Let M be a connected open n-dimensional smooth manifold. Take the one-point-compactification of M, it will be denoted by M=M∪{*}, where * is the infinite point. Since M has countable basis, M is metrizable and {*} is a closed subset and a G_δset. By Urysohn lemma, there exists a continuous function f:→[O,1] with f~-1(O)=M and f~-1(1)=*.Choose a sui-     

The Webster Scalar Curvature and Sharp Upper and Lower Bounds for the First Positive Eigenvalue of the Kohn-Laplacian on Real Hypersurfaces

Let(M, θ) be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue λ_1 of the Kohn-Laplacian □_b on(M, θ). In the present paper, we give a sharp upper bound for λ_1, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when M is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webster scalar curvature, we give a lower bound for λ_1 when the pseudohermitian structure θ is volume-normalized.     

Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds

Let （M,g） be a complete non-compact Riemannian manifold with the m- dimensional Bakry-Emery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation ut = △u - △↓ φ· △ ↓u - aulogu- bu,where φ is a C^2 function, and a ≠ 0 and b are two real constants. This work generalizes the results of Souplet and Zhang （Bull. London Math. Soc., 38 （2006）, pp. 1045-1053） and Wu （Preprint, 2008）.     

The first eigenfunctions and eigenvalue of the p-Laplacian on Finsler manifolds

This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C~(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed eigenvalue of the p-Laplacian on a compact Finsler manifold with nonnegative weighted Ricci curvature,on which a lower bound of the first Dirichlet eigenvalue of the p-Laplacian is also obtained.     

RESEARCH ANNOUNCEMENTS ON "UPPER BOUND ESTIMATES OF EIGENVALUES FOR HORMANDER OPERATORS ON NON-EQUIREGULAR SUB-RIEMANNIAN MANIFOLDS"

1 Introduction and Main Results For n≥2,let M be a n-dimensional connected smooth manifold.We take a fixed smooth measure on M with strictly positive density,and write dx as the measure when we integrate.Also,we simply denote the measure of a set E?M by|E|.     

The distribution of the periodic orbits with open deviation in C~(1+α)non-uniformly hyperbolic systems

Let f : M → M be a C~(1+α) diffeomorphism on a smooth compact Riemannian manifold M and Λ be a Pesin set associated with the ergodic hyperbolic measure μ. Then f : Λ→Λ forms a non-uniformly hyperbolic system. We concern with the distribution of the periodic orbits whose time averages are apart from the space average of μ. Finally, we derive a large deviation result for these periodic orbits with open deviation property.     

Weinstein Conjecture and Holomorphic Spheres

The Weinstein conjecture is one of the central problems in the symplectic geo-metry. Let (M,ω) be a symplectic manifold. Given a compact smooth bypersurfaceS(M,ω) we denote by (S) the collection of all periodic Hamiltonian trajectoriesou S. In [6] A. Weinstein proposed the following conjecture. Conjecture If S is a compact bypersurface of contact type with H~1(S;R) =     

The Degree of Symmetry of Certain Compact Smooth Manifolds Ⅱ

We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2 × V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.     

Classification of manifolds with weakly 1/4-pinched curvatures

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R ² has non-negative isotropic curvature. The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS-0604960.     

The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Suppose that D is a bounded domain with a piecewise C^1 smooth boundary in C^n. Let ψ∈C^1 α(δD). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Boehner-Martinelli kernel, which has integral density ψ. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauehy boundary value problem with the Boehner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.     

A Bayesian approach to the estimation of maps between Riemannian manifolds. II: Examples

Let gJ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space E ^s , and let γ be a smooth map of gJ into a Riemannian manifold Λ. An unknown state θ ∈ gJ is observed via X = θ + ɛξ, where ɛ > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on gJ and smooth estimators g(X) of the map γ we have derived a second-order asymptotic expansion for the related Bayesian risk [3]. In this paper, we apply this technique to a variety of examples. The second part examines the first-order conditions for equality-constrained regression problems. The geometric tools that are utilized in [3] are naturally applicable to these regression problems.     

The X-Ray Transform for a Generic Family of Curves and Weights

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T ^* M. Second author was partly supported by NSF Grant DMS-0400869; third author was partly supported by NSF and a Walker Family Endowed Professorship.     

Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.     

A Bayesian approach to the estimation of maps between Riemannian manifolds

Let Θ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space E ^s, and let γ be a smooth map of Θ into a Riemannian manifold Λ. An unknown state θ ∈ Θ is observed via X = θ + εξ, where ε > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on Θ and smooth estimators g(X) of the map γ we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θ and Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γ is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.      

Proper actions of lattices on contractible manifolds

Every lattice Γ in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if Γ acts on a contractible manifold W and if either?1) the action is properly discontinuous, or?2) W is equipped with a complete Riemannian metric, the action is by isometries and with unbounded orbits, G is simple with finite center and rank >1,?then dimW≥dimG/K. Oblatum 19-I-2001 & 24-IV-2002?Published online: 5 September 2002 RID="*" ID="*"The authors gratefully acknowledge support from the National Science Foundation.     

Construction of non-linear semi-groups using product formulas

Under certain circumstances, the Trotter-Lie formulaW _t=lim(U _t/nV_t/n) ⁿ is used to construct a non-linear semi-groupW _t on closed subsets ofL ^P, 1≦p<∞. In particular we consider the situation whereU _t=e ^tA is a positivity preservingC ₀ (linear) semi-group andV _t is generated by a (non-linear) functionF with certain monotonicity properties. In general,A andF are “singular” onL ^p and no requirement is made that one of them be “relatively bounded” with respect to the other. The generator of the resulting semi-groupW _t turns out to be an extension ofA +F restricted to a suitable domain. Research supported by a Danforth Graduate Fellowship and a Weizmann Postdoctoral Fellowship.     

Scalar curvature,non-abelian group actions,and the degree of symmetry of exotic spheres

It is proved that if a compact manifold admits a smooth action by a compact, connected, non-abelian Lie group, then it admits a metric of positive scalar curvature. This result is used to prove that if ∑ ⁿ is an exoticn-sphere which does not bound a spin manifold, then the only possible compact connected transformation groups of ∑ ⁿ are tori of dimension ≤[(n+1)/2]. Research partially supported by the Sloan Foundation and NSF Grant GP-34785X.