Minimal H\"older regularity implying finiteness of integral Menger curvature

TL;DR

This paper establishes that minimal Hölder regularity guarantees finite integral Menger curvature for curves and manifolds, with the critical Hölder exponent proven optimal through explicit counterexamples.

Contribution

It proves that Hölder continuity with a specific exponent ensures finiteness of integral curvature functionals, extending understanding of geometric regularity conditions.

Findings

Hölder regularity implies finite integral curvature for p > m(m+1)

The critical Hölder exponent is proven optimal with counterexamples

Results apply to both curves in R^3 and m-dimensional manifolds in R^n

Abstract

We study two families of integral functionals indexed by a real number $p > 0$. One family is defined for 1-dimensional curves in $\R^3$ and the other one is defined for $m$-dimensional manifolds in $\R^n$. These functionals are described as integrals of appropriate integrands (strongly related to the Menger curvature) raised to power $p$. Given $p > m(m+1)$ we prove that $C^{1,\alpha}$ regularity of the set (a curve or a manifold), with $\alpha > \alpha_0 = 1 - \frac{m(m+1)}p$ implies finiteness of both curvature functionals ($m=1$ in the case of curves). We also show that $\alpha_0$ is optimal by constructing examples of $C^{1,\alpha_0}$ functions with graphs of infinite integral curvature.