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Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on 3-manifolds. (English) Zbl 0737.53009

From the author’s introduction: “Let \(F\) be a smooth positive function on the boundary of the unit ball in a Euclidean 3-space \(R^ 3\). Then \(F\) defines a functional on immersed surfaces \(M\) in \(R^ 3\) by the formula \[ F(M)=\int_ M F(\nu)dA, \] where \(\nu(x)\) is the unit normal to \(M\) at \(x\) and the integration is with respect to surface area. We show: Theorem 3.4 (condensed version). If \(F\) is even and elliptic, and \(S\) is a smooth simple closed curve on the boundary of a convex set in \(R^ 3\), then for each \(g\geq 0\) there exists a smooth embedded surface that minimizes \(F(M)\) among all embedded surfaces \(M\) with boundary \(\partial M=S\) and genus\((M)\leq g\). Here “\(F\) is even” means that \(F(\nu)\equiv F(-\nu)\), i.e., that \(F(M)\) does not depend on the orientation of \(M\). Ellipticity of \(F\) means that the set \[ \{x: | x| F(x/| x|)\leq 1\} \] is uniformly convex; this is equivalent to ellipticity of the Euler-Lagrange equations for the corresponding functional on graphs.”
Indeed an — in several directions — more general theorem is proved. It is brought in relation to the historical development of the two- dimensional theory of the calculus of variations during the last sixty years.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q10 Optimization of shapes other than minimal surfaces
49Q20 Variational problems in a geometric measure-theoretic setting
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