• Communications of the Korean Mathematical Society
• /
• v.37 no.3
• /
• pp.881-891
• /
• 2022

In the present paper, we have studied Miao-Tam equation on three dimensional almost coKähler manifolds. We have also proved that there does not exist non-trivial solution of Miao-Tam equation on the said manifolds if the dimension is greater than three. Also we give an example to verify the deduced results.

• Journal of the Korean Mathematical Society
• /
• v.55 no.3
• /
• pp.735-747
• /
• 2018

Let M be a real hypersurface of a complex space form with constant curvature c. In this paper, we study the hypersurface M admitting Miao-Tam critical metric, i.e., the induced metric g on M satisfies the equation: $-({\Delta}_g{\lambda})g+{\nabla}^2_g{\lambda}-{\lambda}Ric=g$, where ${\lambda}$ is a smooth function on M. At first, for the case where M is Hopf, c = 0 and $c{\neq}0$ are considered respectively. For the non-Hopf case, we prove that the ruled real hypersurfaces of non-flat complex space forms do not admit Miao-Tam critical metrics. Finally, it is proved that a compact hypersurface of a complex Euclidean space admitting Miao-Tam critical metric with ${\lambda}$ > 0 or ${\lambda}$ < 0 is a sphere and a compact hypersurface of a non-flat complex space form does not exist such a critical metric.

• Kyungpook Mathematical Journal
• /
• v.60 no.1
• /
• pp.177-183
• /
• 2020

The object of the present paper is to study the critical point equation (CPE) on 3-dimensional α-cosymplectic manifolds. We prove that if a 3-dimensional connected α-cosymplectic manifold satisfies the Miao-Tam critical point equation, then the manifold is of constant sectional curvature -α², provided Dλ ≠ (ξλ)ξ. We also give several interesting corollaries of the main result.