73 research outputs found
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 , , 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 of record rank relative to the genus of . Our
first main result is that for any integer with , there exist infinitely many genus hyperelliptic curves over
with at least -points and Mordell--Weil rank
over . Our second main theorem is that if
is an odd prime and contains the -th roots of unity, then there exist
infinitely many genus hyperelliptic curves over with Mordell--Weil rank
at least over .Comment: Comments appreciated
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