Solving Liouville-type Problems on Manifolds with Poincar\'{e}-Sobolev Inequality by Broadening $q$-Energy from Finite to Infinite

Lina Wu

Journal of Mathematics Research, 2017, vol. 9, issue 4, 1-10

Abstract: The aim of this article is to investigate Liouville-type problems on complete non-compact Riemannian manifolds with Poincar\'{e}-Sobolev Inequality. Two significant technical breakthroughs are demonstrated in research findings. The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures varying among negative values, zero, and positive values. Poincar\'{e}-Sobolev Inequality has been applied to overcome difficulties of an extension on manifolds. Poincar\'{e}-Sobolev Inequality has offered a special structure on curved manifolds with a mix of Ricci curvature signs. The second breakthrough is a generalization of $q$-energy from finite to infinite. At this point, a technique of $p$-balanced growth has been introduced to overcome difficulties of broadening from finite $q$-energy in $L^q$ spaces to infinite $q$-energy in non-$L^q$ spaces. An innovative computational method and new estimation techniques are illustrated. At the end of this article, Liouville-type results including vanishing properties for differential forms and constancy properties for differential maps have been verified on manifolds with Poincar\'{e}-Sobolev Inequality approaching to infinite $q$-energy growth.

Keywords: closed differential forms; finite $q$-energy in $L^q$ spaces; H\"{o}lder Inequality; $p$-balanced growth; $p$-harmonic maps; Poincar\'{e}-Sobolev Inequality; $p$-pseudo-coclosed differential forms; Weitzenb\"{o}ck Bochner Formula (search for similar items in EconPapers)
Date: 2017
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