Isoperimetric inequalities involving generalized mean curvature

On minimal immersions in Finsler space

“…al. and Winklmann [9,11,38,39], we establish in the following the convex hull property formulated in Theorem 1.4, the isoperimetric inequalities in Theorem 1.5, and the existence results for Finsler-minimal immersions in Theorem 1.6.…”

“…The first variation formula for general Cartan functionals derived by Räwer [30] and Clarenz [8] were used by Clarenz et al and Winklmann to derive various isoperimetric inequalities for critical immersions of such functionals [10,11,38]. Some of these results are used in the present context in connection with Sect.…”

“…Notice that Winklmann does not use positivity of his Cartan integrand, although we have it here for A F (Z ) automatically by means of (3.3). Now, Winklmann's isoperimetric inequality in [38,Corollary 2.3] reads in our context as…”

“…For part (ii), we benefit from our analysis in the proof of part (i) and use [38,Theorem 3.2] to obtain similarly to (3.11) the preliminary Finsler area estimate…”

Projectability and uniqueness of F-stable immersions with partially free boundaries

“…Obviously, any minimizer X ∈ Ꮿ( , ) of Ᏺ is F-stable, but the converse is not true in general. Standard computations (see [Clarenz 2002, Section 1] or [Winklmann 2002, Proposition 2.1], for example) show that the first variation for X ∈ Ꮿ( , ) is…”