NSF PAR Search | NSF Public Access Repository

https://doi.org/10.1007/s00526-025-02949-z

Mazurowski, Liam; Zhou, Xin (May 2025, Calculus of Variations and Partial Differential Equations)

Abstract We define the half-volume spectrum$$\{{\tilde{\omega }_p\}_{p\in \mathbb {N}}}$$

{{\tilde{ω}}_{p}}_{p \in N}

of a closed manifold$$(M^{n+1},g)$$

(M^{n + 1}, g)

. This is analogous to the usual volume spectrum ofM, except that we restrict top-sweepouts whose slices each enclose half the volume ofM. We prove that the Weyl law continues to hold for the half-volume spectrum. We define an analogous half-volume spectrum$$\tilde{c}(p)$$

\tilde{c} (p)

in the phase transition setting. Moreover, for$$3 \le n+1 \le 7$$

3 \leq n + 1 \leq 7

, we use the Allen–Cahn min-max theory to show that each$$\tilde{c}(p)$$

\tilde{c} (p)

is achieved by a constant mean curvature surface enclosing half the volume ofMplus a (possibly empty) collection of minimal surfaces with even multiplicities.

Search for: All records