IMPA - O Instituto de Matemática Pura e Aplicada

We make use of elementary spectral invariants given by the max-min energy of pseudoholomorphic curves, recently defined by Michael Hutchings, to study periodic $3$-dimensional Reeb flows. We prove that Zoll contact forms on $S^3$ are characterized by $c_1 = c_2 = \mathcal{A}_{\min}$. This follows from the spectral gap closing bound property and a computation of ECH spectral invariants for Zoll contact forms defined on Lens spaces $L(p,1)$ for $p\geq 1$. The former characterization fails for Lens spaces $L(p,1)$ with $p>1$. Nevertheless, we characterize Zoll contact forms on $L(p,1)$ in terms of ECH spectral invariants. We note that a characterization of Besse contact forms also holds for elementary spectral invariants analogously to the one obtained by Dan Cristofaro-Gardiner and Marco Mazzucchelli. This is joint work with Rafael Fernandes.

In 1986, H. Wente constructed a closed immersed torus in $\mathbb{R}^3$ with constant non-zero mean curvature and with one family of spherical curvature lines, disproving a conjecture of H. Hopf. In this work, we produce a family of singly periodic minimal Wente torus with ends in $\mathbb{R}^3$ which are analogous to the singly periodic Riemann minimal surfaces foliated by lines and circles contained in parallel planes. As an application, we produce new examples of free boundary and capillary minimal annuli in two spheres of different centers.

The Lagrange spectrum is the set $L$ of possible finite values of the best approximation constants $k(\alpha)=\limsup_{|p|,|q|\to \infty}|q(q\alpha-p)|^{-1}$, where $\alpha\in \mathbb{R}\setminus \mathbb{Q}$. It is a classical result that the pairs $(p,q)$ which approximate this value must come from convergents $(p_n,q_n)$ of the continued fraction of $\alpha$, and therefore $k(\alpha)=\limsup_{n\to\infty}|q_n(q_n\alpha-p_n)|^{-1}$. Moreira proved that the function $d(t)=HD(L\cap(-\infty,t))$ is a continuous function. The second Lagrange spectra are defined similarly to the classical Lagrange spectrum, but they are related to the problem of finding the best approximations of an irrational number $\alpha$ by rational numbers that are not convergents of the continued fraction of $\alpha$. There are two possible definitions for such a spectrum, because one can consider that a fraction $\frac{p}{q}, (p,q)=k(p_n,q_n),k\geq 2$ that is equivalent to a convergent counts or not as a non convergent. Following this, we define the two second Lagrange spectra $L_2$ and $L_2^*$ and prove that the function $d_2(t)=HD(L_2\cap (-\infty,t))$ is continuous, but $d_2^*(t)=HD(L_2^*\cap (-\infty,t))$ is discontinuous and takes only the values 0 and 1, with a discontinuity at $t=2/3$.

We also study the the spectrum $L_0$ of approximations of complex numbers with real part equal to $1/2$ by Gaussian rationals, which comes from approximating complex numbers with real part equal to $1/2$ by quotients of Gaussian integers. The interesting parts of this spectrum can be realized as a dynamical Lagrange spectrum associated to a real two dimensional horseshoe. We use this description to prove that the function $d_{0}(t)=HD(L_{0}\cap (-\infty,t))$ which measures the Hausdorff dimensions of the intersections of $L_0$ with half-lines is a continuous function of the right endpoint $t$ of the half-line. It follows from results by A. Schmidt that $d_0(2)=0$. We will prove that $d_0(2+\varepsilon)>0$ for every $\varepsilon>0$ and $d_0(2.1)=1$, which implies analogous results for the spectrum $L_{\mathbb{Q}(i)}$ of approximations of all complex numbers by Gaussian rationals.

This is a joint work with Harold Erazo, Carlos Gustavo Moreira and Thiago Vasconcelos.

Nesta palestra exibimos uma família simples de transformações expansoras que nos fornecem uma família de expansões para números no intervalo (0,1) que possuem propriedades similares à expansão em fração contínua. De posse destas apresentaremos algumas questões relacionadas a possíveis explosões nos espectros dinâmicos de Lagrange e Markov introduzidos por C. G. Moreira, como por exemplo a inexistência de interior nos espectros para um membro da família de expansoras. Este é um trabalho em andamento com Cicero Calheiros Filho (UFAL) e Sérgio Romaña (Sun Yat-sen University, China). A palestra está baseada em problemas muito recentes então esperem mais dúvidas do que respostas.

The notion of stability for constant mean curvature hypersurfaces is closely connected to the classical isoperimetric problem and also has physical motivations. This talk concerns the problem of classification for stable CMC hypersurfaces in space forms. We will survey the known results in low-dimensional space forms and present our contributions to this problem in dimension six.

We address a question of do Carmo in six-dimensional Riemannian manifolds with bounded curvature, extending results from lower dimensions. In particular, we show that every complete, finite-index, non-minimal CMC hypersurface immersed in a closed Riemannian manifold with nonnegative sectional curvature is compact.

We also study the general case of a Riemannian manifold with bounded curvature and derive partial results. In particular, we show that a complete, finite-index CMC hypersurface immersed in the hyperbolic space $\mathbb{H}^6$ with mean curvature $|H|>7$ is compact. This gives a partial answer to a question posed by Chodosh in his survey for the ICM.