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Existence results for the nonlinear Hodge minimal surface energy

  • Daniel AgressEmail author
Article
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Abstract

Given a compact Riemannian manifold \((M^n,g)\) and a fixed cohomology class, \([\alpha ^*] \in H^k(M)\), we consider the existence of a minimizer \(\alpha \in [\alpha ^*]\) of the generalized minimal surface energy \(\int _M \sqrt{1+|\alpha |^2} dV_g\). When \(k = 1\), we prove the existence of unique minimizers for every cohomology class \([\alpha ^*]\). Next, when \(k > 1\), we construct examples of singular solutions for finite cohomology class \([\alpha ^*] \in H^k(S^k \times S^k,g)\), where g is conformal to the standard metric on \(S^k \times S^k\). Additionally, we show that when \(k=2\), these singular solutions are also solutions to the Born Infeld equation.

Mathematics Subject Classification

58J05 35J50 49Q05 

Notes

Acknowledgements

The author would like to thank his advisors Patrick Guidotti and Jeffrey Streets, as well as Richard Schoen for their insight and advice.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California IrvineIrvineUSA

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