Polyharmonic hypersurfaces into pseudo-Riemannian space forms
Volker Branding, Stefano Montaldo, Cezar Oniciuc, Andrea Ratto
Abstract
Abstract In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form $$N^{m+1}_t(c)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>N</mml:mi> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>c</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of dimension $$m+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and index t ( $$m\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and $$1 \le t\le m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> ). We shall study hypersurfaces $${M}^{m}_{t'}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:msup> <mml:mi>t</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mi>m</mml:mi> </mml:msubsup> </mml:math> which are polyharmonic of order r (briefly, r -harmonic), where $$r\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> and either $$t'=t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>t</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> or $$t'=t-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>t</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>t</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Let A denote the shape operator of $${M}^{m}_{t'}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:msup> <mml:mi>t</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mi>m</mml:mi> </mml:msubsup> </mml:math> . Under the assumptions that $${M}^{m}_{t'}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:msup> <mml:mi>t</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mi>m</mml:mi> </mml:msubsup> </mml:math> is CMC and $${{\rm{Tr}}}\,{A}^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Tr</mml:mi> <mml:mspace/> <mml:msup> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> is a constant, we shall obtain the general condition which determines that $${M}^{m}_{t'}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:msup> <mml:mi>t</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mi>m</mml:mi> </mml:msubsup> </mml:math> is r -harmonic. As a first application, we shall deduce the existence of several new families of proper r -harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper r -harmonic hypersurfaces ( $$r \ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> ). Finally, we shall obtain the complete classification of proper r -harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form.