Several Metric Properties of Level Curves

TL;DR

This paper derives simple identities connecting metric properties of level curves, such as length and curvature, to the gradient field of functions, with applications in geometry and analysis.

Contribution

It introduces new identities linking level curve metrics to gradients, enhancing understanding of geometric and analytic properties of functions.

Findings

Relations between level curve length and curvature

Connections between gradient fields and level curve metrics

Applications in geometric and analytic contexts

Abstract

This article establishes several remarkably simple identities relating certain metric invariants of level curves of real and complex functions. In particular, we relate lengths of level curves to their curvature and to the gradient field of the function. Some geometric and analytic applications of the results are shown.