arXiv subjects

Norihisa Ikoma

Publications and source records attributed to Norihisa Ikoma.

21 records · Page 2Linked to original sources

arXiv2017

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: Minimization

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Möbius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and analyze the sharp asymptotic behavior of degenerating tori under the action of the Möbius group. In this first work we prove two existence results by minimizing or maximizing a suitable reduced functional, in particular we obtain embedded area-constrained Willmore tori (or, equivalently, toroidal critical points of the Hawking mass under area-constraint) in compact 3-manifolds with constant scalar curvature and in the double Schwarzschild space. In a forthcoming paper new existence theorems will be achieved via Morse theory.

math.DG↗

arXiv2016

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 Möbius 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 Möbius 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.

math.DG↗

arXiv2016

Existence of solutions of scalar field equations with fractional operator

In this paper, the existence of least energy solution and infinitely many solutions is proved for the equation $(1-Δ)^αu = f(u)$ in $\mathbf{R}^N$ where $0<α<1$, $N \geq 2$ and $f(s)$ is a Berestycki-Lions type nonlinearity. The characterization of the least energy by the mountain pass value is also considered and the existence of optimal path is shown. Finally, exploiting these results, the existence of positive solution for the equation $(1-Δ)^αu = f(x,u)$ in $\mathbf{R}^N$ is established under suitable conditions on $f(x,s)$.

math.AP↗