Variational properties of the total inverse mean curvature in the plane under boundary constraints

TL;DR

This work investigates the variational problem for the total inverse mean curvature $\mathcal{F}(\gamma)=\int_\gamma \frac{1}{H}\,d\gamma$ for strictly positively curved planar curves in the half-plane with prescribed boundary and enclosed area. The authors derive the first variation and Euler–Lagrange equation $2+\Delta^\gamma(H^{-2})=\lambda$, construct a unique family of symmetric critical curves $\gamma(x_0,L)$ under the constraint $3x_0<L$, and establish a sharp area–length threshold $\mathcal{A}_0>\frac{3}{2}\pi x_0^2$ for the existence of such critical points, together with a precise parametrization. A thorough second-variation analysis yields a Hardy-type inequality with a positive constant $\mu_{\mathcal{W}_1}>1$, ensuring strong stability of the penalized functional $\mathcal{F}-\lambda\mathcal{A}$, and, correspondingly, stability of $\mathcal{F}$ under area-preserving variations. The paper also proves a local minimality result: the equilibrium configurations are minimizers against small normal perturbations, providing insight into constrained-geometry isoperimetric-type behavior in the plane and offering counterexamples to unconstrained Heintze–Karcher-type optimality under boundary constraints.

Abstract

We study the variational behavior of the total inverse mean curvature of curves with prescribed boundary in the half-plane. We characterize the existence of critical points with prescribed area. We show that such critical points are strongly stable. As an application, we prove a local minimality property.

Figures (1)

• Figure 1: Two equilibrium configurations with same boundary condition and different enclosed area. The configuration on the left-hand side is closer to the treshold \ref{['intro_treshold']}

Theorems & Definitions (52)

• Theorem 1.1
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Lemma 3.1
• proof
• Example 3.2
• Example 3.3