Minimal piecewise linear cones in $\mathbb{R}^4$


  • Asgeir Valfells



We consider three dimensional piecewise linear cones in $\mathbb{R}^4$ that are mass minimal with respect to Lipschitz maps in the sense of [Almgren, F., Mem. Amer. Math. Soc. 4 (1976), no. 165] as in [Taylor, J. E., Ann. of Math. (2) 103 (1976), no. 3, 489–539]. There are three that arise naturally by taking products of $\mathbb{R}$ with lower dimensional cases and earlier literature has demonstrated the existence of two with 0-dimensional singularities. We classify all possible candidates and demonstrate that there are no piecewise linear minimizers outside these five.


Almgren, F., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165.

Blokhuis, A., Brouwer, A., Cohen, A., and Buset, D., The locally icosahedral graphs, Finite geometries (Winnipeg, Man., 1984), 19–22, Lecture Notes in Pure and Appl. Math., 103, Dekker, New York, 1985.

Brakke, K. A., Minimal cones on hypercubes, J. Geom. Anal. 1 (1991), no. 4, 329–338.

Brakke, K. A., The surface evolver, Experiment. Math. 1 (1992), no. 2, 141–165.

Fleming, W. H., On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962), 69–90

Heppes, A., Isogonale sphärische Netze, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 7 (1964), 41–48.

Lawlor, G., and Morgan, F., Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms., Pacific J. Math. 166 (1994), no. 1, 55–83.

Taylor, J. E., The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489–539.



How to Cite

Valfells, A. (2024). Minimal piecewise linear cones in $\mathbb{R}^4$. MATHEMATICA SCANDINAVICA, 130(1).