Asian Journal of Mathematics

Volume 19 (2015)

Number 1

Warped product rigidity

Pages: 135 – 170

DOI: https://dx.doi.org/10.4310/AJM.2015.v19.n1.a6

Authors

Chenxu He (Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania, U.S.A.)

Peter Petersen (Department of Mathematics,University of California at Los Angeles)

William Wylie (Department of Mathematics, Syracuse University, Syracuse, New York, U.S.A.)

Abstract

In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. We obtain a uniqueness result for prescribing the Ricci curvature of a warped product manifold over a fixed base. As an application, this warped product structure will be used to study warped product Einstein structures in [HPW4].

Keywords

warped product, Ricci curvature, Hessian equations, overdetermined linear system of differential equations

2010 Mathematics Subject Classification

53B20, 53C30

Published 12 February 2015