Abstract:
In the 70's Almgren noticed that for a harmonic real-valued function defined on a ball, its L²-norm over a sub-sphere will have an increasing logarithmic derivative with respect to the radius of the mentioned sphere. The increasing of the logarithmic derivative implies the unique continuation, and such inequalities are often referred to "quantitative unique continuation". We examined similar integrals over a more general class of parameterized surfaces by studying harmonic functions defined on compact subdomains of Riemannian manifolds. The integrals over spheres are also generalized to level sets of a given function satisfying certain conditions. If we consider the L²-norms over these level sets parametrized by a generalization of the radius, we again reproduce Almgren’s convexity property. We will sketch the proof of this result and illustrate the usefulness of the convexity result by examining some explicit parameterized families of surfaces, e.g. geodesic spheres and ellipses.