Defining Curvature as a Measure via Gauss–Bonnet on Certain Singular Surfaces
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We show how to define curvature as a measure using the Gauss–Bonnet Theorem on a family of singular surfaces obtained by gluing together smooth surfaces along boundary curves. We find an explicit formula for the curvature measure as a sum of three types of measures: absolutely continuous measures, measures supported on singular curves, and discrete measures supported on singular points. We discuss the spectral asymptotics of the Laplacian on these surfaces.
Mathematics Subject Classification53A05
- 1.Dima, I., Popp, R., Strichartz, R.S.: A convex surface with fractal curvature (in preparation)Google Scholar
- 3.Ivrii, V.: Precise spectral asymptotics for elliptic operators. Lecture Notes in Math, vol. 1100. Springer, Berlin (1984)Google Scholar
- 7.Murray, T., Strichartz, R.: Spectral asymptotics of the Laplacian on surfaces of constant curvature. Commun. Pure Appl. Anal. (to appear)Google Scholar