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1. CJM 2012 (vol 65 pp. 927)
Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$ We consider the following prescribed boundary mean curvature problem
in $ \mathbb B^N$ with the Euclidean metric:
\[
\begin{cases}
\displaystyle -\Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N,
\\[2ex]
\displaystyle \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} \widetilde K(x) u^{2^\#-1} \quad & \text{on }\mathbb S^{N-1},
\end{cases}
\]
where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb
S^{N-1}, 2^\#=\frac{2(N-1)}{N-2}$.
We show that if $\widetilde K(x)$ has a local maximum point,
then the above problem has infinitely many positive solutions
that are not rotationally symmetric on $\mathbb S^{N-1}$.
Keywords:infinitely many solutions, prescribed boundary mean curvature, variational reduction Categories:35J25, 35J65, 35J67 |
2. CJM 2011 (vol 64 pp. 44)
Surfaces of Rotation with Constant Mean Curvature in the Direction of a Unitary Normal Vector Field in a Randers Space |
Surfaces of Rotation with Constant Mean Curvature in the Direction of a Unitary Normal Vector Field in a Randers Space We consider the Randers space $(V^n,F_b)$ obtained by perturbing the Euclidean metric by a translation, $F_b=\alpha+\beta$, where $\alpha$ is the Euclidean metric and $\beta$ is a $1$-form with norm $b$, $0\leq b\lt 1$. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces $(V^3,F_b)$ of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when $b=0$. It also reduces to the equation that characterizes the minimal rotational surfaces in $(V^3,F_b)$ when $H=0$, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for $0\lt b\lt \frac{\sqrt{3}}{3}$. Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical system and by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.
Keywords:Finsler spaces, Randers spaces, mean curvature, Liapunov functions Category:53C20 |
3. CJM 2009 (vol 61 pp. 641)
Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions |
Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form
$\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we
show that if the mean curvature vector of $M^n$ is parallel and the
sectional curvature $K$ of $M^n$ satisfies some inequality, then
the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is
parallel and our manifold $M^n$ is a space form.
Keywords:space forms, parallel isometric immersions, isotropic immersions, totally umbilic, Veronese manifolds, sectional curvatures, parallel mean curvature vector Categories:53C40, 53C42 |