Minimal immersions of compact bordered Riemann surfaces with free boundary
Minimal immersions of compact bordered Riemann surfaces with free boundary
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DOI：
10.1090/s0002-9947-2014-05990-4
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发表时间：
2012-09
影响因子：
1.3
通讯作者：
Jingyi Chen;A. Fraser;Chao Pang
中科院分区：
文献类型：
--
作者：
Jingyi Chen;A. Fraser;Chao Pang
设N是维数大于2的完备齐次正则黎曼流形，M是N的紧致子流形。设$\Sigma$是一个带边界的紧致可定向曲面。我们证明了对于任何连续的f：（\Sigma，\partial \Sigma）\rightarrow（N，M），如果在某些基本群上的诱导同态是单射的，则存在$\Sigma$的分支极小浸入，它解决了自由边界问题$（\Sigma，\partial \Sigma）\rightarrow（N，M）$，并且在所有映射中极小化了与f在基本群上诱导相同作用的面积.进一步，在关于3-流形N的曲率的非负性假设和关于N的边界M的凸性假设下，得到了求解低指标自由边界问题的任一紧致双边极小曲面的亏格、边界分支数和面积的界.
Let N be a complete, homogeneously regular Riemannian manifold of dimension greater than 2 and let M be a compact submanifold of N. Let $\Sigma$ be a compact orientable surface with boundary. We show that for any continuous $f: (\Sigma, \partial \Sigma) \rightarrow (N, M)$ for which the induced homomorphism on certain fundamental groups is injective, there exists a branched minimal immersion of $\Sigma$ solving the free boundary problem $(\Sigma, \partial \Sigma) \rightarrow (N, M)$, and minimizing area among all maps which induce the same action on the fundamental groups as f. Furthermore, under certain nonnegativity assumptions on the curvature of a 3-manifold N and convexity assumptions on M which is the boundary of N, we derive bounds on the genus, number of boundary components and area of any compact two-sided minimal surface solving the free boundary problem with low index.