Flipping Heegaard splittings and minimal surfaces
We show that the number of genus $g$ embedded minimal surfaces in $\mathbb{S}^3$ tends to infinity as $g\rightarrow\infty$. The surfaces we construct resemble doublings of the Clifford torus with curvature blowing up along torus knots as $g\rightarrow\infty$, and arise from a two-parameter min-max s...
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Zusammenfassung: | We show that the number of genus $g$ embedded minimal surfaces in
$\mathbb{S}^3$ tends to infinity as $g\rightarrow\infty$. The surfaces we
construct resemble doublings of the Clifford torus with curvature blowing up
along torus knots as $g\rightarrow\infty$, and arise from a two-parameter
min-max scheme in lens spaces. More generally, by stabilizing and flipping
Heegaard foliations we produce index at most $2$ minimal surfaces with
controlled topological type in arbitrary Riemannian three-manifolds. |
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DOI: | 10.48550/arxiv.2211.03745 |