We prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the mean curvature.

A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this article we construct DUE diffeomorphisms supported on closed manifolds different from tori, providing some counterexamples to a conjecture proposed by Forni in “On the Greenfield-Wallach and Katok conjectures in dimension three,” Contemporary Mathematics 469 (2008).

In this article, we investigate some of the relations between the spectrum of a non-compact, extrinsically bounded submanifold $\varphi : M^m \to N^n$ and the Hausdorff dimension of its limit set $\lim \varphi$. In particular, we prove that if $\varphi : M^2 \to \mathbb{R}^3$ is a (complete) minimal surface immersed into an open, bounded, strictly convex subset $\Omega$ with $C^3$-boundary, then $M$ has discrete spectrum, provided that $\mathcal{H}_{\Psi} (\lim \varphi \cap \Omega) = 0$, where $\mathcal{H}_{\Psi}$ is the Hausdorff measure of order $\Psi(t) = t^2 \lvert \log t \rvert$. Our main theorem, Thm. 2.4, applies to a number of examples recently constructed, by many authors, in the light of Nadirashvili’s discovery of complete bounded minimal disks in $\mathbb{R}^3$, as well as to solutions of Plateau’s problems for non-rectifiable Jordan curves, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures.

On the other hand, we present a simple criterion, called the ball property, whose fulfilment guarantees the existence of elements in the essential spectrum. As an application, we show that some of the examples of Jorge-Xavier and Rosenberg-Toubiana of complete minimal surfaces between two planes have essential spectrum $\sigma_{\mathrm{ess}}(-\Delta) = [0, \infty)$.

A previous result of the authors with Chaput and Perrin states that the closure of the union of all rational curves of fixed degree passing through a Schubert variety in a homogeneous space $G/P$ is again a Schubert variety. In this paper we identify this Schubert variety explicitly in terms of the Hecke product of Weyl group elements. We apply our result to give an explicit formula for any two-point Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula and its equivariant generalization.We also recover a formula for the minimal degree of a rational curve between two given points in a cominuscule variety.

We show that the space of nonpositively curved metrics of a closed negatively curved Riemannian $n$-manifold, $n \geq 10$, is highly non-connected.

We prove that any complete metric on $\mathbb{R}^3$ minus an open ball, with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.