12 research outputs found

    Critical Point Equation on three-dimensional manifolds and the Besse Conjecture

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    In this paper, we present the resolution of the Besse conjecture on a three dimensional compact manifold. We also prove the rigidity of the Miao-Tam critical metric on a three dimensional compact manifold with a smooth boundary

    A complex-angle rotation and geometric complementarity in fermion mixing

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    The mixing among flavors in quarks or leptons in terms of a single rotation angle is defined such that three flavor eigenvectors are transformed into three mass eigenvectors by a single rotation about a common axis. We propose that a geometric complementarity condition exists between the complex angle of quarks and that of leptons in C2\mathbb{C}^2 space. The complementarity constraint has its rise in quark-lepton unification and is reduced to the correlation among θ12,θ23,θ13\theta_{12}, \theta_{23}, \theta_{13} and the CP phase δ\delta. The CP phase turns out to have a non-trivial dependence on all the other angles. We will show that further precise measurements in real angles can narrow down the allowed region of δ\delta. In comparison with other complementarity schemes, this geometric one can avoid the problem of the θ13\theta_{13} exception and can naturally keep the lepton basis being independent of quark basis.Comment: 9 pages, 4 figures, APPC 10 Pohan