| The Gauss-Bonnet formula connects curvature in geometry with characteristic classes in topology.S.-S.Chern[7]gave an intrinsic proof of the Gauss-Bonnet formula on an orientable even-dimensional closed Riemann manifold by considering the Euler-Poincare characteristic as a topological invariant of the tangent sphere bundle to the manifold.M.Atiyah and C.Lebrun[1]proved the Gauss-Bonnet formula and the signature theorem for compact Riemannian 4-manifolds with edge-cone singularities along an embedded 2-manifold.C.T.McMullen[20]proved the corresponding version of the Gauss-Bonnet formula for a special class of stratified spaces,called cone manifolds.Applying the Green’s formula,M.Troyanov[24]proved the Gauss-Bonnet formula for the cone metric on a compact Riemann surface X,where he assumed that the Gaussian curvature K regarding the cone metric can be extended to be a H?lder continuous function on X.We further generalize the work of M.Troyanov,proving that the formula still holds when the Gaussian curvature is Lebesgue integrable under the cone metric.Our theorem and M.Troyanov’s theorem are conceptually similar,but technically very different.The proof idea of both us and M.Troyanov is to first consider the difference between the curvature form of the cone metric and the curvature form of the smooth conformal metric,then by calculation,it is found to be an exact two-form outside the set of cone singularities,and finally,it is reduced to proving that the integral of this exact two-form outside the set of cone singularities equals the contribution of the cone angle using the classical Gauss-Bonnet formula.The difference between our proof and that of Troyanov is:when the Gaussian curvature K can be Holder continuously extended to X,M.Troyanov proved the pointwise gradient estimate of the conformal factor of the cone metric near each cone point;when the Gaussian curvature is Lebesgue integrable under the cone metric,we proved the gradient integral estimate of the conformal factor of the cone metric near each cone point.In the end,both we and Troyanov reach the same conclusion using our respective estimates,proving that the integral of that exact two-form equals the contribution of the cone angle.We also constructed an example of a cone metric whose Gaussian curvature is not integrable.Using the positivity of the function r+r3 sin1/r on the interval(0,δ)and the oscillatory nature of its second derivative,we constructed a rotationally symmetric cone metric with the cone point at the origin in the polar coordinate system,whose Gaussian curvature is not integrable under the cone metric,and the cone angle can take any positive number.Finally,when the Gaussian curvature function Kis bounded,we used Brezis-Merle analysis to prove that the conformal factor of the cone metric can be Holder continuously extended to the cone point.Since the conformal factor satisfies a semi-linear elliptic equation Δu=-K|z|2βe2u,we split the equation into two elliptic boundary value problems,using the Brezis-Merle analysis,Lp regularity theorem,and Sobolev embedding theorem,to obtain the Holder continuity of the conformal factor of the cone metric at the cone point. |
