Weyl Curves and Zeta Determinants of Conic Laplacians on Riemann Surfaces

Abstract

We study self-adjoint realizations of conic Laplacians on compact Riemann surfaces with radial conic metrics whose cone angles are integral multiples of \(2\pi\). Their critical asymptotic coefficients span a finite-dimensional space with a nondegenerate skew-Hermitian Green form, whose Lagrangian subspaces parametrize the self-adjoint realizations. We construct the associated Weyl functions and derive Kre\u{\i}n's resolvent formula, a resolvent trace identity, and, under explicit zeta-regularity hypotheses, a comparison formula for positive-spectrum zeta determinants. The boundary data of formal solutions define a holomorphic Weyl curve in the Grassmannian, with the Weyl functions as its local graph coordinates. Its real restriction is a positive curve in the Lagrangian Grassmannian, and its tangent form is identified with the \(L^2\)-inner product through the Poisson operator. We further identify the boundary determinants with the transition functions of the determinant line bundle induced by the Weyl curve.