论文标题
关于封闭的大地测量和地球网的等均分布
On the equidistribution of closed geodesics and geodesic nets
论文作者
论文摘要
我们表明，鉴于封闭的$ n $ - manifold $ m $，对于一套通用的Riemannian指标$ g $上的$ m $，如果$ n = 2 $，则存在一系列封闭的大地测量学序列，该序列在$ m $中等分为$ m $；以及如果$ n = 3 $，则是嵌入式固定地球网的嵌入式固定地球网的序列。我们使用的主要工具之一是$ 1 $ CYCLES的WEYL LAW，由Liokumovich，Marques和Neves以$ n = 2 $的形式证明，Guth和Liokumovich以$ n = 3 $证明。我们表明，我们可以在任何维度$ n \ geq 2 $上概括地测量网等分的证据，以$ 1 $ -CYCLES的$ N $ -N $ -MANIFOLDS持有，为Weyl Law提供。
We show that given a closed $n$-manifold $M$, for a generic set of Riemannian metrics $g$ on $M$ there exists a sequence of closed geodesics that are equidistributed in $M$ if $n=2$; and an equidistributed sequence of embedded stationary geodesic nets if $n=3$. One of the main tools that we use is the Weyl Law for the volume spectrum for $1$-cycles, proved by Liokumovich, Marques and Neves for $n=2$ and more recently by Guth and Liokumovich for $n=3$. We show that our proof of the equidistribution of geodesic nets can be generalized for any dimension $n\geq 2$ provided the Weyl Law for $1$-cycles in $n$-manifolds holds.
