Free boundary minimal surfaces with three boundary components
The symmetry and topology of a free boundary minimal surface in the Euclidean unit ball do not determine the surface uniquely. There exist infinitely many pairs of non-isometric free boundary minimal surfaces having the same genus , three boundary components and antiprismatic symmetry group of order .
References
- A. Carlotto, M. B. Schulz and D. Wiygul, Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group, Memoirs of the AMS (to appear).
- N. Kapouleas and M. Li, Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc, J. Reine Angew. Math. 776 (2021), 201–254.
genus 2 with 3 boundary components
genus 2 with 3 boundary components
genus 4 with 3 boundary components
genus 4 with 3 boundary components
genus 7 with 3 boundary components
genus 7 with 3 boundary components
genus 11 with 3 boundary components
genus 11 with 3 boundary components
genus 15 with 3 boundary components
genus 15 with 3 boundary components
genus 19 with 3 boundary components
genus 19 with 3 boundary components
genus 27 with 3 boundary components
genus 27 with 3 boundary components
Free boundary minimal surfaces with four boundary components via doubling the critical catenoid
A doubling of the the critical catenoid can be glued to an embedded free boundary minimal surface with sufficiently large, odd genus and four boundary components.
References
- N. Kapouleas and P. McGrath, Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers. Camb. J. Math. 11 (2023).
genus 9 with 4 boundary components
genus 11 with 4 boundary components
genus 13 with 4 boundary components
Free boundary analogues of the higher order Costa–Wohlgemuth surfaces
Conjecturally, the Euclidean unit ball also contains free boundary minimal surfaces which look like the higher order Costa–Wohlgemuth surfaces, i.e. they have sufficiently large, even genus and four boundary components.
genus 6 with 4 boundary components
genus 8 with 4 boundary components
genus 10 with 4 boundary components