In this dissertation, we investigate estimates for the areas of surfaces immersed in 3- dimensional Riemannian manifolds with positive scalar curvature. In particular, we explore the techniques presented by D. Stern (see [Ste22]), which utilize the level sets of a nontrivial harmonic map u : M → S^1. These techniques are employed to establish a rigidity result related to a systolic inequality. Subsequently, we adapt this approach to the context of functions u : M → R that are solutions of a Poisson equation with a non-increasing potential. Through this adaptation, we derive a new systolic inequality that preserves the same spirit as the one presented by D. Stern.

The present work involves the study of the article "Quadratic Response of Random and Deterministic Dynamical Systems" [1], which investigates sufficient conditions for dynamic systems to exhibit a linear response. Additionally, if, in addition to such conditions, other conditions are also observed, the system may exhibit a quadratic response in its stationary measures. It is noteworthy that, although the original article also deals with random dynamical systems, the focus of the current work is on the study of deterministic dynamical systems, emphatically on the study of deterministic dynamical systems with expanding applications on the unit circle. This constitutes a wide range of deterministic dynamical systems and, as we will see, they manifest the desired conditions and, therefore, exhibit linear and/or quadratic responses in their invariant measures when subjected to small perturbations.

The aim of this work is to study the classical Bochner Tubular Theorem from a more modern perspective.

The classic Bochner Tubular Theorem is presented in the classical theory of holomorphic functions and provides a sufficient condition for the extension of holomorphic functions defined on tubular sets in $\mathbb{C}^m$, i.e., sets of the form $U\times\mathbb{R}^m$, where $U$ is an open, connected, non-empty subset of $\mathbb{R}^m$. We present the ideas from the paper \cite{JH2}, which suggest an alternative proof for the Bochner Tubular Theorem. For a complete understanding of this proof, we need to establish some notions and results about convexity, analytic discs, and the Baouendi-Treves Approximation Formula. We only need the particular case of the Baouendi-Treves Formula for the structure generated by the Cauchy-Riemann operators. However, to make the text more comprehensive, we will first present it for arbitrary structures and then for the desired structure (emphasizing that the convergence of the approximate formula can be obtained in other topologies, but in this text it is sufficient to consider only the uniform convergence).

We will also present a microlocal version of the Bochner Tubular Theorem. In this version, we will use a class of FBI transforms (introduced in \cite{BH12}), the notion of analytic wavefront set, and a relation between these concepts. The ideas in the second part of this work are in the paper \cite{SB} written by S. Berhanu. It is important to observe that using this second version, we can extend holomorphic functions defined on sets more general than tubular sets in $\mathbb{C}^m$.

Is to explore the relation between Gevrey regularity and the FBI transform (see 2.1), generalizing known results from the real analytic case (s = 1). Considering par tial differential operators in the form of sums of squares of vector fields, we study the problem of Gevrey hypoellipticity. Our main result is for the class of operators L=\partial_x^2+x^{2(p-1)}\partial_{t_1}^2+x^{2(q-1)}\partial_{t_2}^2 (com 1\leq p \leq q). We show that L is G^s-hypoelliptic in some neighborhood of 0 if and only if s \geq q/p. This monograph follows the M Christ's work from 1997.

This dissertation addresses the study of control for the transport equation, the linearized Korteweg-de Vries equation, and the nonlinear Korteweg-de Vries equation in bounded domains. The aim of this work is to prove the controllability of these systems through the Hilbert’s uniqueness principle.

In this work, we present a classification of the arbitrary dimension and codimension umbilic submanifolds of Sn × R, based on the work of Bruno Mendonça and Ruy Tojeiro (2013). The classification that will be presented extends the classification of umbilical surfaces in S2 × R by Souam and Toubiana (2009), as well as the local description of umbilical hypersurfaces in Sn × R by Van der Veken and Vrancken (2008), to a codimension greater than one. In both works, the umbilical surfaces and the umbilical hypersurfaces are rotational. It will be shown that the submanifolds studied in this work are also rotational and we will present explicit parameterizations.

In this dissertation, we investigate geometric properties of extremal domains, bounded or not. The extremal domains are the domains that are critical points for the first eigenvalue functional for volume-preserving variations.
We show that these domains are characterized by admitting a non-trivial solution to an overdetermined Serrin-type problem.
This motivates us to define the f-extremal domains, when we use an arbitrary function f as the nonlinearity of the overdetermined problem.
The main tool used is the maximum principle in the format of the moving planes method and the main result is the Ros-Sicbaldi Theorem on the proof of Berestycki-Caffarelli-Nirenberg Conjecture in dimension two when the nonlinearity grows at least as a function linear.

In this dissertation, we present some results on index estimates for properly embedded free boundary minimal hypersurfaces in strictly mean convex domains of Euclidean space. The estimates described in this work were obtained by L. Ambrozio, A. Carlotto and B. Sharp in [2], in which they showed that the index of a minimal hypersurface with the properties described above is bounded from below by a linear function of the dimension of its first relative homology group. In three-dimensional ambients, the index of a free boundary minimal surface is bounded from below by a linear function of its genus and the number of boundary components.

The theory of minimal surfaces came up with a problem proposed by Lagrange, which consisted

of the following: given a closed curve without self-intersections, find the surface of

smallest area that has that curve as boundary. Such a problem became known as

the Plateau Problem. It took about 16 years from Lagrange's work to discover non-trivial

examples of minimal surfaces due to Meusnier. The theory was stagnant for 60 years until

Scherk found new examples of minimal surfaces. With the work of Weierstrass it was possible to

obtain more examples of these surfaces. Thereafter there were major developments in theory,

becoming one of the most fertile fields of Differential Geometry. One class of problems studied

is that of estimate the volume of minimal submanifolds immersed in certain ambient

manifolds, such as spheres, hyperplanes, projective spaces, etc. The objective of the present

work is to provide lower bounds of volume of minimal compact submanifolds immersed

in certain symmetrical spaces of rank 1, namely: the unitary sphere Sn, and the real projective space RPn,

complex projective space CPn and quaternionic projective space HPn. It will be shown that if Mm is a

minimal submanifold of Sn, then volM >=V_c(n;M) where V_c(n;M) is the n-conformal volume

of M. Another estimate for this is volM>=c_n, where c_n = vol(Sn). In the case of

M being immersed in projective spaces, we have the lower bounds: c_n/2 in RPm, c_{n+1}/2\pi in

CPm and c_{n+2}=2\pi^2 in HPm.

This thesis aims to present a proof of a theorem of Sebastián Montiel and Antonio Ros which characterizes the compact hypersurfaces, without boundary, with constant r-mean curvature embedded in the space forms. This theorem generalizes the classical theorem of Alexandrov and states that

"The only compact, without boundary, hypersurfaces with constant r-mean curvature for some r=1,...,n, embedded in the Euclidean space, in the open hemisphere of the Euclidean sphere, or in the hyperbolic space, are the geodesic hyperspheres."

We remark that, despite we follow the ideas of Montiel and Ros, we give a new approach for some steps of the proof presented here.

Neste trabalho fazemos um estudo de estabilidade global das soluções especiais, chamadas
solitons, com respeito ao problema de Cauchy (PVI) associado à equação não-linear de Korteweg-de
Vries generalizada (gKdV), para p = 2; 3; 4.
Para a prova do resultado de estabilidade, usamos a abordagem de Weinstein que foi revisada
recentemente no artigo de Muñoz.