Minimal Surfaces with Stratified Branching Sets
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arXiv
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| Format: | Preprint |
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2026
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| author | Franceschini, Federico Mazzeo, Rafe Minter, Paul |
| author_facet | Franceschini, Federico Mazzeo, Rafe Minter, Paul |
| contents | Inspired by the Taubes-Wu construction of $\mathcal{C}^{1,α}$ two-valued harmonic functions by the use of symmetry, we construct minimal surfaces with stratified branching sets as graphs of $\mathcal{C}^{1,α}$ two-valued functions. We give three constructions. The first is perturbative and produces branched minimal submanifolds in arbitrary codimension as two-valued graphs over the $n$-ball or, slightly more generally, over the product of $B^n$ with a torus $\mathbb T^N$, parametrized by boundary data which is required to be small in a suitable norm. The second uses barrier methods together with a reflection argument to produce branched stable minimal hypersurfaces, again as two-valued graphs over the unit $n$-ball or $B^n \times \mathbb T^N$, parametrized by boundary data which now can be large. Finally, using bifurcation theory, we produce compact minimal submanifolds with similarly stratified branching sets in an ambient space $S^n \times \mathbb R$ with a suitable (analytic) warped product metric. These examples give minimal submanifolds with novel frequency values and whose branching sets have non-trivial deeper strata. While the main constructions are fairly elementary, they rely on the use of precisely tailored (and somewhat non-standard) function spaces, combined with a regularity theory which provides full asymptotic expansions around the branching sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27168 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimal Surfaces with Stratified Branching Sets Franceschini, Federico Mazzeo, Rafe Minter, Paul Differential Geometry Analysis of PDEs Inspired by the Taubes-Wu construction of $\mathcal{C}^{1,α}$ two-valued harmonic functions by the use of symmetry, we construct minimal surfaces with stratified branching sets as graphs of $\mathcal{C}^{1,α}$ two-valued functions. We give three constructions. The first is perturbative and produces branched minimal submanifolds in arbitrary codimension as two-valued graphs over the $n$-ball or, slightly more generally, over the product of $B^n$ with a torus $\mathbb T^N$, parametrized by boundary data which is required to be small in a suitable norm. The second uses barrier methods together with a reflection argument to produce branched stable minimal hypersurfaces, again as two-valued graphs over the unit $n$-ball or $B^n \times \mathbb T^N$, parametrized by boundary data which now can be large. Finally, using bifurcation theory, we produce compact minimal submanifolds with similarly stratified branching sets in an ambient space $S^n \times \mathbb R$ with a suitable (analytic) warped product metric. These examples give minimal submanifolds with novel frequency values and whose branching sets have non-trivial deeper strata. While the main constructions are fairly elementary, they rely on the use of precisely tailored (and somewhat non-standard) function spaces, combined with a regularity theory which provides full asymptotic expansions around the branching sets. |
| title | Minimal Surfaces with Stratified Branching Sets |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2603.27168 |