Abstract:
For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.

Abstract:
We compute lower bounds for the Morse index and nullity of constant mean curvature tori of revolution in the three-dimensional unit sphere. In particular, all such tori have index at least five, with index growing at least linearly with respect to the number of the surfaces' bulges, and the index of such tori can be arbitrarily large.

Abstract:
In this work we give a new lower bound on the Morse index for constant mean curvature tori of revolution immersed in the three-sphere $\mathbb{S}^3$, by computing some explicit negative eigenvalues for the corresponding Jacobi operator.

Abstract:
A new approach is proposed for study structure and properties of the total squared mean curvature $W$ of surfaces in ${\bf R}^3$. It is based on the generalized Weierstrass formulae for inducing surfaces. The quantity $W$ (Willmore functional) is shown to be invariant under the modified Novikov--Veselov hierarchy of integrable flows. The $1+1$--dimensional case and, in particular, Willmore tori of revolution, are studied in details. The Willmore conjecture is proved for the mKDV--invariant Willmore tori.

Abstract:
We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions. These theorems provide accurate numerical methods for finding the spectra of those operators over either type of function space. As an application, we numerically compute the Morse index of constant mean curvature tori of revolution in the unit 3-sphere $\mathbb{S}^3$, confirming that every such torus has Morse index at least five, and showing that other known lower bounds for this Morse index are close to optimal.

Abstract:
The paper is devoted to study the Dirichelet energy of moving frames on 2-dimensional tori immersed in the euclidean $3\leq m$-dimensional space. This functional, called Frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. As first result, a Willmore-conjecture type lower bound is established : namely for every torus immersed in $\R^m$, $m\geq 3$, and any moving frame on it, the frame energy is at least $2\pi^2$ and equalty holds if and only if $m\geq 4$, the immersion is the standard Clifford torus (up to rotations and dilations), and the frame is the flat one. Smootheness of the critical points of the frame energy is proved after the discovery of hidden conservation laws and, as application, the minimization of the Frame energy in regular homotopy classes of immersed tori in $\R^3$ is performed.

Abstract:
We consider conformal immersions $f: T^2\rightarrow \mathbb{R}^3$ with the property that $H^2 f^*g_{\mathbb{R}^3}$ is a flat metric. These so called Dirac tori have the property that its Willmore energy is uniformly distributed over the surface and can be obtained using spin transformations of the plane by eigenvectors of the standard Dirac operator for a fixed eigenvalue. We classify Dirac tori and determine the conformal classes realized by them. We want to note that the spinors of Dirac tori satisfies the same system of PDE's as the differential of Hamiltonian stationary Lagrangian tori in $\mathbb{R}^4$. These were classified in [5] .

Abstract:
The energy-critical defocusing nonlinear Schr\"odinger equation on 3-dimensional rectangular tori is considered. We prove that the global well-posedness result for the standard torus of Ionescu and Pausader extends to this class of manifolds, namely, for any initial data in $H^1$ the solution exists globally in time.

Abstract:
We discuss the energy level splitting $\Delta\epsilon$ due to quantum tunneling between congruent tori in phase space. In analytic cases, it is well known that $\Delta\epsilon$ decays faster than power of $\hbar$ in the semi-classical limit. This is not true in non-smooth cases, specifically, when the tori are connected by line on which the Hamiltonian is not smooth. Under the assumption that the non-smoothness depends only upon the x- or p-coordinate, the leading term in the semi-classical expansion of $\Delta\epsilon$ is derived, which shows that $\Delta \epsilon$ decays as $\hbar^{k+1}$ when $\hbar\to 0$ with k being the order of non-smoothness.

Abstract:
The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean curvature tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue estimates on Riemannian spin bundles in dimension 2 are given.