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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Locally constrained inverse curvature flows


Authors: Julian Scheuer and Chao Xia
Journal: Trans. Amer. Math. Soc. 372 (2019), 6771-6803
MSC (2010): Primary 53C21, 53C24, 53C44
DOI: https://doi.org/10.1090/tran/7949
Published electronically: August 28, 2019
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Abstract: We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order--namely, the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to deduce a new Minkowski-type inequality in the anti-de Sitter Schwarzschild manifolds and a weighted isoperimetric-type inequality in hyperbolic space.


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Additional Information

Julian Scheuer
Affiliation: Albert-Ludwigs-Universität, Mathematisches Institut, Abteilung Reine Mathematik, Ernst-Zermelo-Stra\normalfont{ß}e 1, 79104 Freiburg, Germany
Email: julian.scheuer@math.uni-freiburg.de

Chao Xia
Affiliation: School of Mathematical Sciences, Xiamen University, 361005 Xiamen, People’s Republic of China
Email: chaoxia@xmu.edu.cn

DOI: https://doi.org/10.1090/tran/7949
Keywords: Inverse curvature flow, constrained curvature flow, geometric inequality, warped product space
Received by editor(s): August 20, 2017
Published electronically: August 28, 2019
Additional Notes: The first author was being supported by the “Deutsche Forschungsgemeinschaft” (DFG, German research foundation), research grant “Quermassintegral preserving local curvature flows”, number SCHE 1879/3-1. The research of the second author was supported in part by NSFC (Grant No. 11501480, 11871406), the Natural Science Foundation of Fujian Province of China (Grant No. 2017J06003) and the Fundamental Research Funds for the Central Universities (Grant No. 20720180009). Part of this work was done while he was visiting the Institute of Mathematics at Albert-Ludwigs-Universität Freiburg. He would like to thank the Institute for its hospitality.
Article copyright: © Copyright 2019 American Mathematical Society