Differential equations on riemannian manifolds and their geometric applications
Differential equations on riemannian manifolds and their geometric applications
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DOI：
10.1002/cpa.3160280303
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发表时间：
1975-05
影响因子：
3
通讯作者：
Shiu-yuen Cheng;S. Yau
中科院分区：
文献类型：
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作者：
Shiu-yuen Cheng;S. Yau
微分几何中的大多数问题都可以归结为黎曼流形上的微分方程问题。我们在这里的主要目的是研究这些方程及其在几何中的应用。在第一节中，我们证明了如果完备黎曼流形的测地球的体积至多像二次多项式那样增长，则在该流形上不存在非平凡的正超调和函数。实际上，获得了更精确的定量结果。我们应用这一点证明了一个伯恩斯坦型定理：如果H是定义在R3上的同号函数，且aH/aX，= aHfaXz= O和aH/aX，SO，则不存在定义在R2上的每个点的平均曲率由H给出的图。在第二节中，我们简化了文[9]中“极大值原理”的一个证明，并给出了一些应用。例如，如果M是完备黎曼流形，使得在紧集外Ricci曲率由（dim M+ &）/r ~ 2有界，其中r是到某个不动点的距离，E> 0是一个固定常数，则M不允许任何非常数正超调和函数.这被定义为M的所有紧子域的第一特征值（对于Dirichlet问题）的无穷大。证明了若f是定义在M上的任意正函数，则M的第一特征值由inf（-Aflf）下界.然后我们观察到，如果M的测地线球的体积多项式增长，M的第一特征值必须为零。这两个事实一起给出了定义在M上的正函数的强限制。研究这些事实的目的之一是完全证明文[8]中研究的下列定理：欧氏空间中任何具有常平均曲率的完备凸超曲面是广义柱面。(The Osserman和Klotz [4]通过完全不同的方法获得了二维版本。）
Most of the problems in differential geometry can be reduced to problems in differential equations on Riemannian manifolds. Our main purpose here is to study these equations and their applications in geometry. In Section 1, we prove that if the volume of the geodesic balls of a complete Riemannian manifold grows at most like a quadratic polynomial, then there is no non-trivial positive superharmonic function defined on this manifold. In fact, a more precise quantitative result is obtained. We apply this to prove a Bernstein type theorem: If H is a function of the same sign defined on R3 with aH/aX,= aHfaXz= O and aH/aX, SO, then there is no graph defined on R2 whose mean curvature at each point is given by H. In Section 2, we simplify a proof of the “maximal principle” in [9] and give some applications. For example, if M is a complete Riemannian manifold such that outside a compact set the Ricci curvature is bounded from below by (dim M+ &)/r2, where r is the distance from some fixed point and E> O is a fixed constant, then M does not admit any non-constant positive superharmonic function.In Section 3, we study the first eigenvalue of a complete (non-compact) Riemannian manifold M. This is defined to be the infinimum of the first eigenvalues (for the Dirichlet problem) of all compact subdomains of M. We prove that if f is any positive function defined on M, the first eigenvalue of M is bounded from below by inf (-Aflf). We then observe that if the volume of the geodesic balls of M grows polynomially, the first eigenvalue of M has to be zero. These two facts together give strong restrictions on positive functions defined on M. One purpose of studying these facts is to give a complete proof of the following theorem studied in [8]: Any complete convex hypersurface with constant mean curvature in euclidean space is a generalized cylinder.(The two-dimensional version was obtained by Osserman and Klotz [4] by a completely different method.)