The Semi-linear Torsional Rigidity on a Complete Riemannian Two-Manifold

TL;DR

This paper investigates properties of the $ ext{[0,1)}$-torsional rigidity on complete Riemannian two-manifolds, revealing new results even in the Euclidean plane, with implications for isoperimetric inequalities and variational formulas.

Contribution

It establishes new properties of semi-linear torsional rigidity on Riemannian surfaces, including optimal isoperimetry, first variation, and monotonicity, extending known results to more general settings.

Findings

Derived new isoperimetric inequalities for torsional rigidity.

Established first variation formulas for the $ ext{[0,1)}$-torsional rigidity.

Proved monotonicity properties on Riemannian two-manifolds.

Abstract

This note is concerned with some essential properties (optimal isoperimetry, first variation, and monotonicity formula) of the so-called $[0,1)\ni\gamma$-torsional rigidity $\mathcal{T}_{\gamma,\mathsf{g}}$ on a complete Riemannian two-manifold $(\mathbb M^2,\mathsf{g})$. Even in the special case of $\mathbb R^2$, major results are new.