Mohammadjavad Habibivostakolaei

We introduce a geometric structure – a conformal Killing–Yano Ricci soliton (CKY–RS) – that couples conformal Ricci soliton (CRS) geometry to conformal Killing–Yano (CKY) 2–forms. The soliton field of the CRS geometry is given by the divergence of the CKY 2–form. We introduce a conserved CKY–Cotton current and derive a compatibility identity relating the Cotton tensor, the CRS obstruction tensor, and the CKY 2–form. In 4–dimensional Lorentzian signature, we show that, under non-degeneracy and closedness assumptions on the CKY form, a CKY–RS structure forces the conformal representative to be locally Kerr–NUT–(A)dS. For a closed non-degenerate CKY on a Kerr–NUT–(A)dS background, the conformal deformation is necessarily trivial. For Einstein backgrounds of arbitrary dimension and signature, the conformal factor satisfies an eigenvalue equation and an Obata–type Hessian equation. If the background is also compact or a CKY orbit is periodic, the conformal factor is an invariant of the CKY–flow and we obtain simple spectral obstructions to non-trivial CKY–RS structures. From the Hessian equation, we obtain obstruction and classification results for the non-trivial conformal sector, including product/Brinkmann geometries and a Weyl–aligned branch. Finally, we give explicit constructions for static spherically symmetric geometries and BTZ backgrounds, including a CKY–RS realization with a time-dependent conformally flat representative. These results provide a geometric framework for studying CRS with hidden symmetry structure, with potential applications to exact geometries in general relativity.

We study the spectral stability of Dirichlet eigenvalues on an embedded annulus whose boundary evolves by curve shortening flow while the ambient surface evolves under the two dimensional Ricci flow using variational formulas, Rellich–type identities, and harmonic capacity methods, we relate eigenvalue variations to geometric deficit and modulus. We establish quantitative bounds comparing the spectrum of the evolving annulus with that of a flat cylinder of equal modulus. As a consequence, we obtain geometric stability and a spectral gap estimate controlled by the deficit functional.