Some Aspects of the Dynamic of V=H−H
Some Aspects of the Dynamic of V=H−H
复制标题
V=H−H 动态的一些方面
DOI：
10.1006/jdeq.1998.3626
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发表时间：
1999
影响因子：
2.4
通讯作者：
G. Fusco
中科院分区：
文献类型：
--
作者：
G. Bellettini;G. Fusco
根据方程V=HΓ(T)考虑曲面−(T)的演化，其中V是Γ(T)的法向速度，H是两个主曲率之和，H是H在Γ(T)上的平均值。我们研究了Γ(T)与固定曲面Σ垂直相交的情形，并在假设Γ(T)与Γ(T)之间的封闭区域的体积很小的情况下讨论了Σ(T)的动力学性质。我们证明了，在这种情况下，如果Γ(0)在半球附近，Γ(T)保持其近半球的形状，并在Σ上沿着∇的两个主曲率的和HΣ的切向梯度Σ的轨道近似滑动。我们还证明了，如果p∈Σ是∇HΣ的非退化零点，且a>；0足够小，则存在一个常平均曲率曲面，它在半径a的半球附近，中心在p附近，并且与Σ正交相交。
Abstract We consider the evolution of a surface Γ ( t ) according to the equation V = H − H , where V is the normal velocity of Γ ( t ), H is the sum of the two principal curvatures and H is the average of H on Γ ( t ). We study the case where Γ ( t ) intersects orthogonally a fixed surface Σ and discuss some aspects of the dynamics of Γ ( t ) under the assumption that the volume of the region enclosed between Γ ( t ) and Σ is small. We show that, in this case, if Γ (0) is near a hemisphere, Γ ( t ) keeps its almost hemispherical shape and slides on Σ crawling approximately along orbits of the tangential gradient ∇ H Σ of the sum H Σ of the two principal curvatures of Σ . We also show that, if p ∈ Σ is a nondegenerate zero of ∇ H Σ and a >0 is sufficiently small, then there is a surface of constant mean curvature which is near a hemisphere of radius a with center near p and intersects Σ orthogonally.