Abstract
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 \((M^3,g,e^{-f}d\mu )\). This result is similar to that proved by Li and Wei (J Geom Anal 25:421–435, 2015). Li and Wei assumed \(Ric_f\ge k>0\), where \(Ric_f\) is the Bakry-Émery Ricci curvature, and that the embedded f-minimal surfaces have fixed genus. Here we suppose \(R_f^P+\frac{1}{2}|\overline{\nabla }f|^2>0\), where \(R_f^P\) is the Perelman scalar curvature, and uniform bound on the f-index of the embedded f-minimal surfaces.
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The authors were partially supported by CNPq/Brazil and FAPEMIG grants.
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Meira, A., Gonçalves, R.A. On the space of f-minimal surfaces with bounded f-index in weighted smooth metric spaces. manuscripta math. 162, 559–563 (2020). https://doi.org/10.1007/s00229-019-01144-7
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DOI: https://doi.org/10.1007/s00229-019-01144-7