Some Rigidity Results Related to the Obata-Type Equation

Abstract

In this paper, we provide some rigidity results for compact manifolds with smooth boundary by using appropriate geometric or topological assumptions and the Obata-type equation

{\mathrm{\nabla }}^{2}f-fg=0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nabla ^2 f -fg =0$$\end{document}

with Robin boundary condition

f_{\nu }=cf

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_{\nu } = cf$$\end{document}

, where

c>1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$c>1$$\end{document}

(note that in this case f has no critical points). By the same idea, we also use equations

{\mathrm{\nabla }}^{2}f+fg=0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nabla ^2 f +fg =0$$\end{document}

and

{\mathrm{\nabla }}^{2}f=0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nabla ^2 f=0$$\end{document}

to establish similar rigidity results. The proofs of our rigidity results are mainly based on the warped product structures on compact manifolds determined by the Obata-type equation (see Proposition 1.11), and we also provide the corresponding structure on complete non-compact manifolds with compact boundary (see Proposition 1.13). It should be pointed that we have actually provided all possible structures determined by the equation

{\mathrm{\nabla }}^{2}f-fg=0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nabla ^2 f -fg =0$$\end{document}

with

f_{\nu }=cf

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_{\nu } = cf$$\end{document}

for complete manifolds with compact boundary.

Keywords

Rigidity result / The Obata-type equation / Robin boundary condition / 53C21 / 53C24

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School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

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