Peer-reviewedCC BYJournal of the European Mathematical Society
Topological control for min-max free boundary minimal surfaces
Abstract
We establish general bounds on the topology of free boundary minimal surfaces obtained via min-max methods in compact, three-dimensional ambient manifolds with mean convex boundary. We prove that the first Betti number is lower semicontinuous along min-max sequences converging in the sense of varifolds to free boundary minimal surfaces. In the orientable case, we obtain an even stronger result which implies that if the number of boundary components increases in the varifold limit, then the genus decreases at least as much. We also present several compelling applications, such as the variational construction of a free boundary minimal trinoid in the Euclidean unit ball.
Related
- Exterior free-boundary minimal surfaces of every topological type with at least two ends
- Free boundary minimal surfaces with connected boundary and arbitrary genus
- Free Boundary Minimal surfaces in the Euclidean Three-Ball close to the boundary
- Inequivalent complexity criteria for free boundary minimal surfaces
- Free boundary minimal surfaces of genus zero
BibTeX
@article{franz2026topological,
title = {{Topological control for min-max free boundary minimal surfaces}},
author = {Giada Franz and Mario Schulz},
journal = {Journal of the European Mathematical Society},
year = {2026},
month = jun,
doi = {10.4171/jems/1800},
eprint = {2307.00941},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05843915v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - Topological control for min-max free boundary minimal surfaces
AU - Giada Franz
AU - Mario Schulz
PY - 2026
DA - 2026/06/18
JO - Journal of the European Mathematical Society
DO - 10.4171/jems/1800
UR - https://hal.science/hal-05843915v1
AB - We establish general bounds on the topology of free boundary minimal surfaces obtained via min-max methods in compact, three-dimensional ambient manifolds with mean convex boundary. We prove that the first Betti number is lower semicontinuous along min-max sequences converging in the sense of varifolds to free boundary minimal surfaces. In the orientable case, we obtain an even stronger result which implies that if the number of boundary components increases in the varifold limit, then the genus decreases at least as much. We also present several compelling applications, such as the variational construction of a free boundary minimal trinoid in the Euclidean unit ball.
SN - 1435-9855
SN - 1435-9863
ER -