Abstract
We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives L' and L'' of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.
Highlights
The geometry of level sets has been a vital and fruitful topic of investigations involving variety of function and space settings
We show that in the smooth setting the isoperimetric inequality (ln L(t)) ≥ 0, t1 ≤ t ≤ t2, (1)
One should recall, for instance, the theory due to Colding and Colding–Minicozzi which applies to non-parabolic manifolds and exhibits monotonicity formulas for integrals overlevel sets of the Green functions of quantities defined in terms of their gradients, [21,22,23]
Summary
The geometry of level sets has been a vital and fruitful topic of investigations involving variety of function and space settings.
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