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In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X , then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to -g as the genus g goes to infinity.

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In this report, we propose a new semi-fragile watermarking algorithm for the authentication of 3D models based on integral invariants. To do so, we embed a watermark image by modifying the integral invariants of some of the vertices. Basically, we shift a vertex and its neighbors in order to change the integral invariants. To extract the watermark, test all the vertices for the embedded information, and combine them to recover the watermark image. How many parts can the watermark image be recovered would help us to make the authentication decision. Experimental test shows that this method is robust against rigid transform and noise attack, and useful to test purposely attack besides transferring noise and geometrical transforming noise. An additional contribution of this paper is a new algorithm for computing two kinds of integral invariants.

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