Minimizing the squared mean curvature integral for surfaces in space forms. (English) Zbl 0778.53001

Hsu, Lucas; Kusner, Rob; Sullivan, John

Exp. Math. 1, No. 3, 191-207 (1992).

The experiments which are described in this paper support the conjecture that smooth minimizers of the squared mean curvature integral for closed surfaces exist for each genus and that they are stereographic projections of certain minimal surfaces in the three-sphere. The authors use a discrete version of this integral for polyhedral surfaces and apply Brakke’s surface evolver [see K. A. Brakke, ibid. 1, No. 2, 141-165 (1992; Zbl 0769.49033)] to compute the evolution of the polyhedral surfaces under the corresponding flow. A detailed description of the results of the experiments is given, including some hints for appropriate adjustments of the numerical procedures making the iterations convergent.

Reviewer: Bernd Wegner (Berlin)

MSC:

53A05	Surfaces in Euclidean and related spaces
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53-04	Software, source code, etc. for problems pertaining to differential geometry

Keywords:

smooth minimizers; squared mean curvature integral; polyhedral surfaces; Brakke’s surface evolver

Citations:

Zbl 0769.49033

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