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The Monge-Ampère equation and warped products of higher rank. (English) Zbl 1133.53030

Summary: We show that a warped product \(M_f=\mathbb R^n\times_f\mathbb R\) has higher rank and nonpositive curvature if and only if \(f\) is a convex solution of the Monge-Ampère equation. In this case we show that \(M\) contains an Euclidean factor.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C24 Rigidity results
35J60 Nonlinear elliptic equations
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References:

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