Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals
Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals
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DOI：
10.1007/bf01388908
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发表时间：
1987-06
影响因子：
3.1
通讯作者：
B. White
中科院分区：
文献类型：
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作者：
B. White
设 N 为三维黎曼流形，go 为每个 xeN 分配实数 go (x, v) 并将切空间中的单位向量 v 分配给 x 处的 N 的函数。然后 go 定义曲面上的泛函如下：(,) go (M)=~ gO (x, v (x)) dx x~ M，其中 v (x) 是在 x 处垂直于 M 的单位，积分是相对于 M 上的表面积。注意，如果 gO 在 gO (x, v)-gO (x,-v) 中是偶数，则即使 M 是不可定向的，(,) 也有意义。如果 M 满足 gO 的欧拉-拉格朗日方程，我们就说 M 是 gO 平稳的。例如，如果 gO (x, v)-1，则 gO (M) 是 M 的面积，并且当且仅当 M 是最小曲面时，M 是 gO 平稳的。在本文中，我们证明了规则gO-平稳表面的曲率估计和紧性定理，假设gO是椭圆形，即如果对于某些2> 0，(gO (x, v/[v [)-2) Iv [是veTan x N的凸函数。例如，我们证明定理0./j gO是偶椭圆被积函数，gO和D2gO是C 2'~，N是紧流形具有严格的 gO-凸边界，并且 M 是一个嵌入的 gO-静止表面，且 OM c ON，则主曲率 oJ" M 受一个常数限制，该常数仅取决于 gO、N、M 的面积和亏格，[[0M [[2,=，以及？ M 的“嵌入性”，即
Let N be a three dimensional riemannian manifold and go a function that assigns a real number go (x, v) to each xeN and unit vector v in the tangent space to N at x. Then go defines a functional on surfaces as follows:(,) go (M)=~ gO (x, v (x)) dx x~ M where v (x) is the unit normal to M at x and the integration is with respect to surface area on M. Note if gO is even in that gO (x, v)-gO (x,-v), then (,) makes sense even if M is non-orientable. We say that M is gO-stationary if it satisfies the Euler-Lagrange equations for gO. For example, if gO (x, v)-1, then gO (M) is the area of M, and M is gO-stationary if and only if M is a minimal surface. In this paper we prove curvature estimates and compactness theorems for regular gO-stationary surfaces provided gO is elliptic, ie if for some 2> 0,(gO (x, v/[v [)-2) Iv [is a convex function of veTan x N. For example we proveTheorem0./j gO is an even elliptic integrand, gO and D2gO are C 2'~, N is a compact maniJold with strictly gO-convex boundary, and M is an embedded gO-stationary surface with OM c ON, then the principal curvatures oJ" M are bounded by a constant depending only on gO, N, the area and genus of M,[[0M [[2,=, and the" embeddedness" of? M, ie