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The catenoid estimate and its geometric applications. (English) Zbl 1439.53064

Given a 3-manifold \(M\) and a closed orientable unstable embedded minimal surface \(\Sigma\) in \(M\), the authors construct a sweepout (i.e., a one-parameter family of surfaces) between the boundary of a tubular neighborhood about \(\Sigma\) and a graph on \(\Sigma\) where each surface in the sweepout has area less than twice that of \(\Sigma\). This catenoid estimate allows them to rule out the phenomenon of multiplicity in min-max theory. As a consequence, they show (1) that the width of a 3-manifold with positive Ricci curvature can be realized by an orientable minimal surface, (2) that minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal, and (3) that the “doublings” of the Clifford torus by N. Kapouleas and S.-D. Yang [Am. J. Math. 132, No. 2, 257–295 (2010; Zbl 1198.53060)] can be constructed variationally by an equivariant min-max procedure.
The authors also obtain a result in higher dimensions which is analogous to (1).

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49K35 Optimality conditions for minimax problems

Citations:

Zbl 1198.53060
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