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- W4308670129 abstract "Given $I,Binmathbb{N}cup {0}$, 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." @default.
- W4308670129 created "2022-11-14" @default.
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- W4308670129 date "2022-11-07" @default.
- W4308670129 modified "2023-09-23" @default.
- W4308670129 title "Geometry of branched minimal surfaces of finite index" @default.
- W4308670129 doi "https://doi.org/10.48550/arxiv.2211.03529" @default.
- W4308670129 hasPublicationYear "2022" @default.
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