Índice do catenoide capilar imerso na bola unitária Euclidiana

Abstract

This thesis consists of several results on minimal capillary hypersurfaces contained in the Euclidean unit ball $\mathbb{B}_{1}^{n+1}(0)$. In the first part, we studied the Morse index of the capillary catenoids contained in the Euclidean ball $\mathbb{B}_{1}^{3}(0)$. We prove that, if $\Sigma_{c}$ is a capillary catenoid contained in $\mathbb{B}_{1}^{3}(0)$ where $c$ is the capillary constant, then the Morse index of $\Sigma_{c}$ is between 3 and 7 ($3\leq\Ind(\Sigma_{c})\leq 7$). Furthermore, we obtain that any capillary catenoid with a contact angle close to $\frac{\pi}{2}$ a Morse index equal to 4, as does the critical catenoid. Also, we verify that there is no capillary catenoid in the unit ball with a contact angle less than $\widetilde{\theta}\approx 1.3421$. To find our estimate of the capillary catenoid index, we focused on the analysis of two simpler eigenvalue problems associated with our minimal capillary surface (Jacobi-Steklov and fixed boundary). In the second part, we show some results of classification of minimal capillary hypersurfaces considering that the coordinate functions of the Gauss map are Jacobi-Steklov eigenfunctions. For $n=2$, we show that, if any of the eigenvalues associated with the coordinate functions of the Gauss map is zero, then the surface is a totally geodesic disk. Still for $n=2$, we prove that, if two eigenvalues associated with the coordinate functions of the Gauss map are equal and different from zero, then the surface is a capillary catenoid. In the last part, we find two integral identities that allow us to present a different proof of the following result that has already been shown by other authors: if $\Sigma^{n}$ is a type-II stable stationary hypersurface(stability in the space of functions that have zero mean on the boundary) on the unit ball, then $\Sigma^{n}$ is a totally geodesic $n$-disk.