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Estimates for parametric elliptic integrands. (English) Zbl 1002.53035

Let \(M\) be a three-dimensional Riemannian manifold and \(\varphi\) a real-valued function defined on the unit sphere bundle \(S(M)\) of \(M\) such that \(\varphi\geq 1\). For an oriented surface \(N\) immersed in \(M\) let \(\Phi\) be a functional defined by \(\Phi(N)=\int_{x\in N}\varphi(x,n(x)) dx\), where \(n(x)\) is the unit normal to \(N\) and \(dx\) is the area measure on \(N\). A surface \(N\) is said to be \(\Phi\)-stationary if it is a critical point for \(\Phi\), and it is \(\Phi\)-stable if its second variation is nonnegative for deformations of compact support. The functional \(\Phi\) is elliptic if there is a \(\lambda>0\) such that, for each \(x\in M\), \(v\to \left(\varphi(x,v/||v||)-\lambda\right)||v||\) is a convex function. In this paper, the authors provide bounds on area and total curvature for intrinsic balls in \(\Phi\)-stable surfaces.

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
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