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On the space of f-minimal surfaces with bounded f-index in weighted smooth metric spaces
manuscripta mathematica ( IF 0.5 ) Pub Date : 2019-08-27 , DOI: 10.1007/s00229-019-01144-7
Adson Meira , Rosivaldo Antonio Gonçalves

In this note we prove a compactness theorem for the space of connected closed embedded f -minimal surfaces, of bounded f -index, in a simply connected smooth metric measure space $$(M^3,g,e^{-f}d\mu )$$ ( M 3 , g , e - f d μ ) . This result is similar to that proved by Li and Wei (J Geom Anal 25:421–435, 2015). Li and Wei assumed $$Ric_f\ge k>0$$ R i c f ≥ k > 0 , where $$Ric_f$$ R i c f is the Bakry-Émery Ricci curvature, and that the embedded f -minimal surfaces have fixed genus. Here we suppose $$R_f^P+\frac{1}{2}|\overline{\nabla }f|^2>0$$ R f P + 1 2 | ∇ ¯ f | 2 > 0 , where $$R_f^P$$ R f P is the Perelman scalar curvature, and uniform bound on the f -index of the embedded f -minimal surfaces.

中文翻译:

在加权光滑度量空间中具有有界 f 指数的 f 极小曲面的空间

在本笔记中,我们证明了一个紧性定理,用于在一个简单连接的平滑度量空间 $$(M^3,g,e^{-f}d \mu )$$ ( M 3 , g , e-fd μ ) 。这个结果与 Li 和 Wei (J Geom Anal 25:421–435, 2015) 证明的结果相似。Li 和 Wei 假设 $$Ric_f\ge k>0$$ R icf ≥ k > 0 ,其中 $$Ric_f$$ R icf 是 Bakry-Émery Ricci 曲率,并且嵌入的 f 极小曲面具有固定的属。这里我们假设 $$R_f^P+\frac{1}{2}|\overline{\nabla }f|^2>0$$ R f P + 1 2 | ∇ ¯ f | 2 > 0 ,其中 $$R_f^P$$ R f P 是 Perelman 标量曲率,并且是嵌入的 f 极小曲面的 f 指数上的均匀边界。
更新日期:2019-08-27
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