Second Jacobi eigenvalues and spectral rigidity for surfaces in Berger spheres

Abstract

In this paper, we establish an upper bound for the second eigenvalue of the scalar Jacobi operator of an arbitrary closed two-sided surface immersed in a contracted Berger sphere. The estimate involves the total squared mean curvature, the Euler characteristic, and an explicit nonpositive term determined by the angle between the surface normal and the Hopf direction. The proof combines the realization of a Berger sphere as a geodesic hypersurface of a complex projective plane, the first standard embedding of the latter into a Euclidean sphere, a weighted Hersch--Li--Yau balancing argument, and the conformal invariance of the Willmore functional. No minimality or constant mean curvature assumption is imposed. As an application, if $1/3\leqα\leq1$ and the surface has nonpositive Euler characteristic, then its second Jacobi eigenvalue is nonpositive; it is strictly negative for $α>1/3$. At the critical value $α=1/3$, equality forces the immersed image to be congruent to the minimal Clifford torus; in the embedded category, this yields a complete characterization of the equality case.