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

TL;DR

This paper derives a straightforward formula for the scalar curvature of level sets in Euclidean spaces, linking it to the gradient and Laplacian of the defining function, with applications to harmonic and p-harmonic functions.

Contribution

It introduces a simple, explicit formula for scalar curvature of level sets in Euclidean spaces, enhancing geometric analysis of harmonic functions.

Findings

Derived a formula relating scalar curvature to gradient and Laplacian.

Applied the formula to analyze low-dimensional p-harmonic functions.

Extended the analysis to high-dimensional harmonic functions.

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.