Minimal surfaces, stability, growth of area, first eigenvalue.

This dissertation is based on the recent results of O. Munteanu, C.-J. Sung, and J. Wang, published in 2023 in the referenced article [MSW23]. Our main motivation lies in the study of the area growth of geodesic balls and estimates for the bottom of the spectrum of the Laplacian operator on stable minimal surfaces in a three-dimensional manifold with scalar curvature bounded from below. After a review of the topics and techniques involved, we initially focus on the case of Euclidean space R3. In this case, we obtain an optimal area estimate that allows us to assert that stable minimal surfaces have area growth exactly like the Euclidean plane. This is sufficient to prove that complete stable minimal surfaces in R3 are planes. This conclusion is already known from the contributions of Do Carmo and Peng [dCP79], Fisher-Colbrie and Schoen [FCS80], and Pogorelov [Pog81]. The technique for proving the area estimate can be adapted to obtain area estimates in the case of a more general ambient manifold.

In the second part of the work, we focus on upper estimates for the bottom of the spectrum of stable minimal hypersurfaces. Initially, we recall that the bottom of the spectrum is closely related to the growth of the volume of geodesic balls. Motivated by this fact, we obtain our estimates using test functions constructed in terms of the Green’s function. Due to technical reasons, these estimates are only valid for stable minimal hypersurfaces in complete manifolds with dimension up to six.