A general existence result for the Toda system on compact surfaces

In this paper we consider the Toda system of equations on a compact surface, which is motivated by the study of models in non-abelian Chern-Simons theory. We prove a general existence result using variational methods. The same analysis applies to a mean field equation which arises in fluid dynamics.Comment: 28 pages, 1 figure, accepted on Advances in Mathematic

A topological join construction and the Toda system on compact surfaces of arbitrary genus

We consider a Toda system of Liouville equations defined on a compact surface which arises as a model for non-abelian Chern-Simons vortices. For the first time the range of parameters $\rho_1 \in (4k\pi , 4(k+1)\pi)$, $k \in \mathbb{N}$, $\rho_2 \in (4\pi, 8\pi )$ is studied with a variational approach on surfaces with arbitrary genus. We provide a general existence result by means of a new improved Moser-Trudinger type inequality and introducing a topological join construction in order to describe the interaction of the two components

Variational aspects of singular Liouville systems

I studied singular Liouville systems on compact surfaces from a variational point of view. I gave sufficient and necessary conditions for the existence of globally minimizing solutions, then I found min-max solutions for some particular systems. Finally, I also gave some non-existence results

On a critical Maxwell equation in nonlocal media

In this paper, we study the existence of solutions for a critical time-harmonic Maxwell equation in nonlocal media. By introducing some suitable Coulomb spaces involving curl operator, we are able to obtain the ground state solutions of the curl-curl equation via the method of constraining Nehari-Pankov manifold. Correspondingly, some sharp constants of the Sobolev-like inequalities with curl operator are obtained by a nonlocal version of the concentration-compactness principle.Comment: 33 pages. arXiv admin note: text overlap with arXiv:2002.00613 by other author

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds II: Morse Theory

This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the M\"obius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory; to this aim we establish new geometric expansions of the derivative of the Willmore functional on exponentiated small Clifford tori degenerating, under the action of the M\"obius group, to small geodesic spheres with a small handle. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.Comment: Final version, to appear in the American Journal of Mathematic

Constructing curves of high rank via composite polynomials

We improve on a construction of Mestre--Shioda to produce some families of curves $X/\mathbb{Q}$ of record rank relative to the genus $g$ of $X$. Our first main result is that for any integer $g \geqslant 8$ with $g \equiv 2 \pmod 3$, there exist infinitely many genus $g$ hyperelliptic curves over $\mathbb{Q}$ with at least $8g+32$ $\mathbb{Q}$-points and Mordell--Weil rank $\geqslant 4g + 15$ over $\mathbb{Q}$. Our second main theorem is that if $g+1$ is an odd prime and $K$ contains the $g+1$-th roots of unity, then there exist infinitely many genus $g$ hyperelliptic curves over $K$ with Mordell--Weil rank at least $6g$ over $K$.Comment: Comments appreciated