A Simple Formula for Scalar Curvature of Level Sets in Euclidean Spaces

Abstract

A simple formula is derived for the Ricci scalar curvature of any smooth level set $\psi (x_0,x_1,...,x_n)=C$ embedded in the Euclidean space ${\mathbb{R}}^{n+1}$, in terms of the gradient $\nabla \psi$ and the Laplacian $\Delta \psi$. Some applications are given to the geometry of low-dimensional $p$-harmonic functions and high-dimensional harmonic functions.

15 pages

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