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<p>We consider a class of functionals that depends on two arguments, whose partial map with respect to one of the arguments achieves its minimum, and for which the Palais–Smale condition is only partially satisfied in the other variable. Using Ekeland’s variational principle we show existence of a global minimizer and we present a new Mountain Pass Theorem. This abstract functional framework can be applied to the geometric question of parametrizing the moduli space of minimal immersions in hyperbolic 3-manifolds by using suitable data. Apart from this example, our setup can also be applied to a wide class of nonlinear PDEs systems involving general elliptic operators.</p>

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A Mountain Pass Theorem and Moduli Space of Minimal Immersions

  • 期刊论文

  • 发表日期：

    2025年3月28日

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  • Marcello Lucia

摘要

We consider a class of functionals that depends on two arguments, whose partial map with respect to one of the arguments achieves its minimum, and for which the Palais–Smale condition is only partially satisfied in the other variable. Using Ekeland’s variational principle we show existence of a global minimizer and we present a new Mountain Pass Theorem. This abstract functional framework can be applied to the geometric question of parametrizing the moduli space of minimal immersions in hyperbolic 3-manifolds by using suitable data. Apart from this example, our setup can also be applied to a wide class of nonlinear PDEs systems involving general elliptic operators.