Spectral gap estimates on compact manifolds-猫眼课题宝

紧流形上的谱间隙估计

DOI：

10.1090/s0002-9947-99-02039-5

发表时间：

1999

期刊：

影响因子：

通讯作者：

中科院分区：

文献类型：

作者：

K. Oden;C. Sung;Jiaping Wang;Jiaping Wang

对于带边界的紧致黎曼流形，它的质量间隙是第一个和第二个最小的Dirichlet本征值之差。本文利用变分方法，得到了紧致流形的质量间隙的一个关于几何量的显式下界估计。0。引言设(M，g)是具有非空边界的紧致黎曼流形，∂M.作用于M上的函数的拉普拉斯-贝尔特拉米算子定义为∆(F)=1√G∂∂i(√GG∂f∂j)，其中M上的度量由DS=gijdx⊗dx，(G)=(Gij)−1和G=det(Gij)给出。Dirichlet边界条件下拉普拉斯算子的本征值是对某些非零本征函数ui满足{∆ui+liui=0，x∈M，ui(X)=0，x∈∂M的常数li。众所周知，Dirichlet问题的谱满足specD(M)={0&lt；L1&lt；L2≤L3··→∞}，因此间隙L2−L1，有时被称为“质量间隙”，是非平凡的。在[S-W-Y-Y]中，通过分析函数φ=U2U1来估计质量间隙。结果表明，∆φ+2∇对数U1·∇对数φ+(L2−L1)φ=0。因此，L2−L1表现为某一偏微分算子的特征值。此外，由于很容易看出φ在M上光滑且满足∂φ∂ν=0，所以L2−L1是某一偏微分算子的Neumann特征值。因此，Li和Yau[L-Y]的梯度估计技巧可以用来证明L2−L1≥π4D2，其中M是凸欧氏区域，D是M的直径。然而，凸性的假设似乎对这一论点至关重要，因为编辑们在1995年8月22日收到了黑森，并在1997年2月13日以修订的形式收到了黑森。1991年数学科目分类。主58C40。C©1999美国数学学会

For a compact Riemannian manifold with boundary, its mass gap is the difference between the first and second smallest Dirichlet eigenvalues. In this paper, taking a variational approach, we obtain an explicit lower bound estimate of the mass gap for any compact manifold in terms of geometric quantities. 0. Introduction Let (M, g) be a compact Riemannian manifold with non-empty boundary, ∂M . The Laplace-Beltrami operator (Laplacian) acting on functions on M is defined by ∆(f) = 1 √ G ∂ ∂i (√ Gg ∂f ∂j ) , where the metric on M is given by ds = gijdx ⊗ dx , (g) = (gij)−1 and G = det(gij). Eigenvalues of the Laplacian under Dirichlet boundary conditions are constants li which satisfy { ∆ui + liui = 0, x ∈ M, ui(x) = 0, x ∈ ∂M, for some nonzero eigenfunctions ui. It is well known that the spectrum of the Dirichlet problem satisfies SpecD(M) = {0 < l1 < l2 ≤ l3 · · · → ∞}, so that the gap l2 − l1, sometimes referred to as the “mass gap” , is nontrivial. In [S-W-Y-Y] the mass gap is estimated by analyzing the function φ = u2 u1 . They show that ∆φ + 2∇ log u1 · ∇ log φ + (l2 − l1)φ = 0. Therefore, l2− l1 appears as an eigenvalue of a certain partial differential operator. Moreover, since it is easy to see that φ is smooth on M and satisfies ∂φ ∂ν = 0, l2−l1 is a Neumann eigenvalue of a certain partial differential operator. Thus, the gradient estimate techniques of Li and Yau [L-Y] can be employed to show that l2 − l1 ≥ π 4D2 where M is a convex Euclidean domain and D is the diameter of M . The assumption of convexity seems to be crucial, however, to the argument since the Hessian Received by the editors August 22, 1995 and, in revised form, February 13, 1997. 1991 Mathematics Subject Classification. Primary 58C40. c ©1999 American Mathematical Society

Spectral gap estimates on compact manifolds

紧流形上的谱间隙估计

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