Minimising curvature. A higher dimensional analogue of the Plateau problem. (English) Zbl 0592.53003
Nonlinear analysis, Miniconf. Canberra/Aust. 1984, Proc. Cent. Math. Anal. Aust. Natl. Univ. 8, 113-122 (1984).
[For the entire collection see Zbl 0552.00008.]
The author summarizes results given in detail in two other papers [Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J. 35, 45-71 (1986; Zbl 0561.53008); \(C^{1,\alpha}\) multiple function regularity and tangent cone behaviour for varifolds with p-summable second fundamental form (to appear)]. Some of the proofs are sketched. The author describes the types of varifolds and convergence theorems which he has used in solving the problem of minimizing the curvature of a k-dimensional surface among all k- dimensional surfaces having a given boundary. The author also describes his regularity theorem for k-dimensional varifolds having p-power summable second fundamental form, \(p>k\).
The author summarizes results given in detail in two other papers [Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J. 35, 45-71 (1986; Zbl 0561.53008); \(C^{1,\alpha}\) multiple function regularity and tangent cone behaviour for varifolds with p-summable second fundamental form (to appear)]. Some of the proofs are sketched. The author describes the types of varifolds and convergence theorems which he has used in solving the problem of minimizing the curvature of a k-dimensional surface among all k- dimensional surfaces having a given boundary. The author also describes his regularity theorem for k-dimensional varifolds having p-power summable second fundamental form, \(p>k\).
Reviewer: H.Parks
MSC:
53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |