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1.
We consider a differential expression ${H=\nabla^*\nabla+V}We consider a differential expression H=?*?+V{H=\nabla^*\nabla+V}, where ?{\nabla} is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufficient condition for H to have an m-accretive realization in the space L p (E), where 1 < p <  +∞. We study the same problem for the operator Δ M  + V in L p (M), where 1 < p < ∞, Δ M is the scalar Laplacian on a complete Riemannian manifold M, and V is a locally integrable function on M.  相似文献   

2.
Let R be a noetherian ring, \mathfraka{\mathfrak{a}} an ideal of R, and M an R-module. We prove that for a finite module M, if Hi\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is minimax for all i ≥ r ≥ 1, then Hi\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is artinian for i ≥ r. A local–global principle for minimax local cohomology modules is shown. If Hi\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is coatomic for i ≤ r (M finite) then Hi\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is finite for i ≤ r. We give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.  相似文献   

3.
Circularm-functions are introduced on smooth manifolds with boundary. We study the distribution of their critical circles and construct an example of a four-dimensional manifoldM 4 with boundary ∂M 4 that satisfies the condition ξ(∂M 4)=ξ(M 4,∂M 4)=0 but does not contain any circularm-function. We prove that a manifold with boundaryM n (n≥5) such that ξ(∂M n , ∂M n )=0 always contains a circularm-function without critical points in the interior manifold. Sukhumi Branch of the Tbilisi University, Sukhumi. Translated from Ukrainskii Matermaticheskii Zhurnal, Vol. 46, No. 6, pp. 776–781, June, 1994.  相似文献   

4.
Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim r → ∞ e2r s(r) = 0, then (M, g) has to be isometric to ℍ n . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim r → ∞ r 2 s(r) = 0, then (M, g) is isometric to ℝ n , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if Ka on a geodesic ball B p (R) in M and K = a on ∂B p (R), then K = a on B p (R).  相似文献   

5.
The purpose of this article is to study compactness of the complex Green operator on CR manifolds of hypersurface type. We introduce (CR-P q ), a potential theoretic condition on (0, q)-forms that generalizes Catlin’s property (P q ) to CR manifolds of arbitrary codimension. We prove that if an embedded CR-manifold of hypersurface type of real dimension at least five satisfies (CR-P q ) and (CR-P n-1-q ), then the complex Green operator is a compact operator on the Sobolev spaces Hs0,q(M){H^s_{0,q}(M)} and Hs0,n-1-q(M){H^s_{0,n-1-q}(M)} , if 1 ≤  q ≤  n−2 and s ≥  0. We use CR-plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle.  相似文献   

6.
LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK H≥4λ and antiholomorphic Ricci curvatureρ A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C P n−1(λ) andM=C P n(λ). Moreover, we give some results on the volume of of tubes aboutP inM. Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01.  相似文献   

7.
An Application of a Mountain Pass Theorem   总被引:3,自引:0,他引:3  
We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, uH 1 0(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L -function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ s 0 f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.  相似文献   

8.
 Let M be a 2m-dimensional compact Riemannian manifold with Anosov geodesic flow. We prove that every closed bounded k form, k≥2, on the universal covering of M is d(bounded). Further, if M is homotopy equivalent to a compact K?hler manifold, then its Euler number χ(M) satisfies (−1) m χ(M)>0. Received: 25 September 2001 / Published Online: 16 October 2002  相似文献   

9.
A thickening of a finite CW-complex X is by definition a compact manifold M of the same simple homotopy type as X. We give a model for the cochain complex of the boundary of that manifold, C *M), as a module over the cochain algebra C *(X). We also show how to construct an algebraic model of the rational homotopy type of δC *(M) from a model of X. Using this rational model, we prove a new formula for the rational Lusternik–Schnirelmann category of X. Received: 24 September 1999  相似文献   

10.
Let G be a bounded open subset in the complex plane and let H~2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1-1 with respect to the Lebesgue measure on D and if the Riemann map belongs to the weak-star closure of the polynomials in H~∞(W). Our main theorem states: in order that for each M∈Lat (M_z), there exist u∈H~∞(G) such that M=∨{uH~2(G)}, it is necessary and sufficient that the following hold: (1) each component of G is a perfectly connected domain; (2) the harmonic measures of the components of G are mutually singular; (3) the set of polynomials is weak-star dense in H~∞(G). Moreover, if G satisfies these conditions, then every M∈Lat (M_z) is of the form uH~2(G), where u∈H~∞(G) and the restriction of u to each of the components of G is either an inner function or zero.  相似文献   

11.
We use the integral geometric formulas in the symplectic space of geodesics of a Riemannian manifold to derive various inequalities of isoperimetric type. We give a sharp lower bound for the area of the minimal bubble spanning a spherical curve in ℝ3. We also present an “inverse Croke inequality” relating the area of the boundary of a complex domain in a Riemannian manifold to the injectivity radius and the volume of the domain. We prove a sharp lower bound for the ground state of the harmonic oscillator operator inL 2(M), whereM is a Hadamard manifold. This article is dedicated to my dear friend Julia Rashba  相似文献   

12.
Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D * is an FC-group, then D is a field. Let M be a maximal subgroup of GL n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)] < ∞.  相似文献   

13.
For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F, we define a morphism of cohomology groups rk: Hk(A,F) → Hk(Lx,Fx), called the localization map, where Lx is the adjoint algebra at x ∈ M. The main result in this paper is that if M is simply connected, or H (LX,FX) is trivial, then T is injective. This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.  相似文献   

14.
We study isoperimetric regions on Riemannian manifolds of the form (M n × (0, π), sin2(t)gdt 2) where g is a metric of positive Ricci curvature ≥ n − 1. When g is an Einstein metric we use this to compute the Yamabe constant of (M ×\mathbbR, g+ dt2 ){(M \times \mathbb{R}, g+ dt^2 )} and so to obtain lower bounds for the Yamabe invariant of M × S 1.  相似文献   

15.
Let M^n be a smooth, compact manifold without boundary, and F0 : M^n→ R^n+1 a smooth immersion which is convex. The one-parameter families F(·, t) : M^n× [0, T) → R^n+1 of hypersurfaces Mt^n= F(·,t)(M^n) satisfy an initial value problem dF/dt (·,t) = -H^k(· ,t)v(· ,t), F(· ,0) = F0(· ), where H is the mean curvature and u(·,t) is the outer unit normal at F(·, t), such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.  相似文献   

16.
Let M be a very ample line bundle on a smooth complex projective variety Y and let ϕ M :YP(H 0(Y, M)*) be the map associated to M; we are concerned with the problem to see whether the syzygies of ϕ M (Y) give information on the syzygies of ϕ M s (Y). In particular we prove that if Y is a smooth complex projective variety and M is a line bundle on Y satisfying Property N p , then M s satisfies Property N p if sp. Received: 11 June 1999 / Revised version: 22 November 1999  相似文献   

17.
Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aMM. Let D(−) ≔ Hom R (−, E) be the Matlis dual functor, where EE(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x 1, …, x n is a regular sequence on M contained in α, then H (x1, …,xnR n D(H a n (M))) is a homomorphic image of D(M), where H b i (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H (x1, …,xn)R n (D(H a n (M)))) ⋟ D(D(M)).  相似文献   

18.
Let M be a quantizable symplectic manifold. If ψt is a loop in the group {Ham}(M) of Hamiltonian symplectomorphisms of M and A is a 2k-cycle in M, we define a symplectic action κA(ψ)∊ U(1) around ψt(A), which is invariant under deformations of ψ, and such that κA(ψ) depends only on the homology class of A. Using properties of κA( ) we determine a lower bound for ♯π1(Ham(O)), where O is a quantizable coadjoint orbit of a compact Lie group. In particular we prove that ♯π1(Ham(CPn)) ≥ n+1. Mathematics Subject Classifications (2000): 53D05, 57S05, 57R17, 57T20.  相似文献   

19.
We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π n (M) →H n (M; ℤ) by the formula degf =f*[S n ], where [S n ] εH n (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH 2(M, ℤ/2ℤ) such thatβ ·w 2(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ 2 α ·w 2(M)=0, whereρ 2: ℤ → ℤ/2ℤ is reduction modulo 2.  相似文献   

20.
Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an (n + 1)-dimensional Riemannian manifold N^n+1 whose (n - 1)th Ricci curvature satisfying Ric^N(n-1) (n - 1)c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If |φ|^2 - (n- 2)√n(n- 1)|H||φ|+ n(2n - 1)(H^2+ c) 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c + H^2 〉 0, then M must have only one end.  相似文献   

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