Geometry of branched minimal surfaces of finite index

TL;DR

This paper studies the geometry and existence of complete finitely branched minimal surfaces in three-dimensional space with bounded Morse index and branching order, deriving scale-invariant estimates and exploring stability and non-orientability.

Contribution

It introduces new scale-invariant weak chord-arc estimates for such surfaces and general intrinsic area monotonicity formulas, extending understanding of their geometric properties.

Findings

Derived scale-invariant weak chord-arc estimates depending on index and branching

Established intrinsic monotonicity of area formulas for submanifolds in Riemannian manifolds

Constructed examples of stable, non-orientable minimal surfaces generalizing Henneberg surfaces

Abstract

Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse index at most $I$ and total branching order at most $B$. Previous works of Fischer-Colbrie and Ros explain that such surfaces are precisely the complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature and finite total branching order. Among other things, we derive scale-invariant weak chord-arc type results for such an $M$ with estimates that are given in terms of $I$ and $B$. In order to obtain some of our main results for these special surfaces, we obtain general intrinsic monotonicity of area formulas for $m$-dimensional submanifolds $\Sigma$ of an $n$-dimensional Riemannian manifold $X$, where these area estimates depend on the geometry of $X$ and upper bounds on the lengths of the mean curvature vectors of $\Sigma$. We also describe a family of complete, finitely branched minimal surfaces in $\mathbb{R}^3$ that are stable and non-orientable; these examples generalize the classical Henneberg minimal surface.