Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds

Abstract We consider the Steklov problem on differential $$p\text {-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mtext>-</mml:mtext> </mml:mrow> </mml:math> forms defined by Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., $$0\text {-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>-</mml:mtext> </mml:mrow> </mml:math> forms), highlighting the influence of the underlying topology on the spectrum for $$p\text {-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mtext>-</mml:mtext> </mml:mrow> </mml:math> forms in general.