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A variational characterization of complex submanifolds

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In this note, we generalize our results in Arezzo and Sun (Reine Angew Math, doi:10.1515/crelle-2013-0097, 2012) to integer p-currents of any degree. We prove that if the mass of a current, as a functional of the ambient metric, has a critical or stable point in some special directions, then the current is complex. This holds for any dimension and codimension. We also study a natural functional on the space of currents representing a fixed homology class, closely related to the first derivative of the Mass in our new approach, detecting the deviation of a surface from being holomorphic.

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Acknowledgments

The first author wishes to thank C. De Lellis for pointing out reference [1] to our attention and him, G. De Philippis and E. Spadaro for many important discussions. He also wishes to thank CIRM-FBK (Trento) for providing an ideal working atmosphere. Part of the work are carried out when the second author was a postdoc at CIRM-FBK (Trento). He thanks the center for their hospitality. Claudio Arezzo was partially supported by FIRB Project RBFR08B2HY. The second author was supported by the National Natural Science Foundation of China, No. 11401440.

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Authors and Affiliations

  1. The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

    Claudio Arezzo

  2. Universitá di Parma, Parma, Italy

    Claudio Arezzo

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Jun Sun

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  1. Claudio Arezzo
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  2. Jun Sun
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Correspondence to Claudio Arezzo.

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Arezzo, C., Sun, J. A variational characterization of complex submanifolds. Math. Ann. 366, 249–277 (2016). https://doi.org/10.1007/s00208-015-1322-9

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  • DOI: https://doi.org/10.1007/s00208-015-1322-9

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